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On þ-adic expansions of algebraic numbers
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On þ-adic expansions of algebraic numbers
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On p-adic Expansions of Algebraic Numbers by Hsing-Hau Chen A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (COMPUTER SCIENCE) DISSERTATION COMMITTEE Ming-Deh Huang (Chair), Leonard M. Adleman, Eric M. Friedlander, and Sheldon Kamienny August 2017 Hsing-Hau Chen to bone of my bones, and flesh of my flesh. ii Acknowledgements Give thanks unto the Lord and bless His name, for guiding the sheep of His pasture through this journey. Praise be to theLord, for leading the sheep to know Him more and fulfill His will. Before the mountains were brought forth, or ever thou hadst formed the earth and the world, even from everlasting to everlasting, thou art God. I would like to express my immense gratitude to my advisor Ming-Deh Huang for his guidance, expertise, patience and support towards my doctoral study. Were it not for working under his tutelage, I would not have seen the light at the end of the tunnel of my doctoral pursuit. I am grateful to Ming-Deh Huang, Leonard Adleman, Eric Friedlander and Sheldon Kamienny for serving on my dissertation committee. I would like to thank Eric and Sheldon especially for their warm regard and encouragement. I am indebted to the faculty of the computer science department at USC, in particular to David Kempe, who not only served on my qualifying exam committee, but also gave various computer science theory courses during my initial years, from which I found my interest in theory. Fellow computer science theory students iii Joseph Bebel, Rui Chao, Yu Cheng, Ho Yee Cheung, Ehsan Emamjomeh-Zadeh, Bailan Li, Lian Liu, Anand Kumar Narayanan, Alana Shine, and Haifeng Xu were a constant source of enthusiasm and inspiration. I was fortunate to learn several foundational mathematics courses through the instructionofEricFriedlander, CharlesLanski, andSheldonKamiennyinthemath- ematics department at USC. But for the generosity of the computer science department in supporting me through teaching assistantships, my research investigations would not have been possible. In particular, Aaron Cote and Shahriar Shamsian were of great help through teaching assistantship offerings for their theory courses. I am thankful to the department graduate advisor Lizsl De Leon Spedding for helping me deal with ease all things administrative. To the body of Christ at CBCWLA I am blessed to be part of, I am thankful. Their company and persistent prayers carry me through tough times. With my whole heart I thank my parents and family, in particular my beloved wife Sandy and three kids Pierrette, Andreas, and Philippus, for their endless love and support that helped me materialize this dissertation. iv Table of Contents Dedication ii Acknowledgements iii Abstract vii Chapter 1 Introduction 1 1.1 Finite or Periodic p-adic Expansion Problem . . . . . . . . . . . . . 2 1.2 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter 2 c-Roundedness 10 2.1 Upper Bounding the Closest Vector . . . . . . . . . . . . . . . . . . 10 2.2 The Lattice Spanned by the Unit Group . . . . . . . . . . . . . . . 16 2.3 c-Rounded Algebraic Numbers . . . . . . . . . . . . . . . . . . . . . 19 Chapter 3 Finite or Periodic p-adic Expansions 24 3.1 Principal Prime p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1 A Constructive Proof . . . . . . . . . . . . . . . . . . . . . . 26 3.1.2 Ring of Fractions Defined by p . . . . . . . . . . . . . . . . . 34 3.1.3 Lifting p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Prime but Not Necessarily Principal P . . . . . . . . . . . . . . . . 37 3.2.1 Reducing the Case of P . . . . . . . . . . . . . . . . . . . . 38 3.2.2 Ring of Fractions Defined by P . . . . . . . . . . . . . . . . 39 3.3 The Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4 An Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Chapter 4 Examples 46 4.1 Number Fields of Unit Rank Zero . . . . . . . . . . . . . . . . . . . 46 4.2 Gaussian Rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2.1 p is ramified . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2.2 p splits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2.3 p is inert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 A Real Quadratic Field . . . . . . . . . . . . . . . . . . . . . . . . . 51 v 4.3.1 p is ramified . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3.2 p splits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3.3 p is inert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4 An Imaginary Quadratic Field with Non-Principal Ideals . . . . . . 54 4.4.1 p is ramified . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.4.2 p splits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.4.3 p is inert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Chapter 5 Conclusion and Future Directions 60 Bibliography 61 vi Abstract It is well known that a p-adic number has a finite or periodic p-adic expansion if and only if it is rational. An interesting question is whether the characterization generalizes to algebraic numbers; that is, whether algebraic numbers in a number fieldK are precisely those elements inK p which have a p-adic expansion that is either finite or eventually periodic, whereK p denotes the local completion ofK at a prime ideal p. To answer this question, we propose a more general notion of p-adic expansion for algebraic numbers, where given a number fieldK and a prime ideal p inK, a different choice of uniformizer forK p fromK is allowed in each stage of the expansion. With the notion ofp-adic expansion, we prove that algebraic numbers in KarepreciselythoseelementsinK p whichhaveafiniteorperiodicp-adicexpansion. On the other hand we call ap-adic expansion a-adic expansion if a uniformizer can be fixed for the entire expansion. As a special case we show that, algebraic numbers in a number fieldK that has unit rank zero are precisely those elements in the local completion ofK at which have a finite or periodic -adic expansion where is a generator for a principal prime ideal. vii Chapter 1 Introduction The concept of p-adic numbers was introduced by Kurt Hensel in 1897 [Hen1] primarily as an attempt to bring the ideas and techniques of power series methods into number theory. By now the influence of p-adic numbers and p-adic analysis has extended far beyond the original scope. It is an independent field of study as well as an integral part of number theory. In the context of computation, some problems and algorithms are p-adic by nature, such as polynomial factorization with Hensel’s Lemma [Hen2] and point countingonhyperellipticcurvesoverfinitefieldswithp-adiccohomology[Ked,Lau]. Completion and deformation techniques with an intensive use of power series and p-adic integers come up in many areas of symbolic and analytic computations, such as polynomial factorization [BGP + ], polynomial or differential system solving [LS], and analytic continuation [GHK + ]. This has motivated the design of relaxed algorithms [BHL] in order to better implementp-adic numbers; in addition, relaxed Hensel lifting for algebraic systems 1 [BL] was developed to compute the Hensel lifting of a root from a residue ring to its p-adic completion. More recently, a general framework allowing a precise study of p-adic precision was designed [CRV1] and applied to certain linear algebraic tasks[CRV2], polynomialfactorization[CRV3], differentialsystemsolving[LV], and computation of characteristic polynomials of p-adic matrices [CRV4]. In addition, the p-adic arithmetic allows error-free representation of fractions and error-free arithmetic using fractions [KRS1, KRS2, Kri, Koç, LLS]. 1.1 Finite or Periodic p-adic Expansion Problem Letb 2 be an integer. Theb-ary expansion of every rational number is either finite or eventually periodic [Bun]. Conversely a real algebraic number has a finite or periodic b-ary expansion only if it is rational. Let p 2 be a prime number. A p-adic expansion of a rational number is a concrete representation of the number as an element in the field Q p of p-adic numbers. Thep-adic expansion of every rational number under thep-adic valuation is also either finite or eventually periodic [Rob]. Conversely a p-adic number has a finite or periodic p-adic expansion only if it is rational. More generally given an algebraic number in an algebraic number fieldK, if p is a prime ideal ofK containing the rational prime p, one can consider a p- adic expansion as a concrete representation of 2K as an element in the local completionK p ofK at p. An interesting question that arises is whether algebraic 2 numbers inK are precisely those elements ofK p which have ap-adic expansion that is either finite or eventually periodic. In this thesis we give an affirmative answer to the question under a sufficiently general notion of p-adic expansion. We also present an algorithm for constructing suchp-adic expansions. Our results generalize the techniques in [CH] for construct- ing finite or periodic p-adic expansions for algebraic integers. We introduce the notion of rounded algebraic numbers and prove that every algebraic number has a rounded associate. This theorem is important for extending our techniques from algebraic integers to algebraic numbers. The theorem is interesting for its own sake and may have further applications as well. As a motivational example to the question concerning p-adic expansions, con- sider the number field, Q () where 3 = 1, the field of Eisenstein rationals. Let = 1. Since 3 = (1) (1 2 ) = (1) 2 (1 +), is a ramified prime above the rational prime 3. Consider writing 3 -adically with respect to a complete set of residues = f0; 1; 2g. Since 3 = (1) 2 (1 +) = 2 (2) = 2 2 3 , writing 3 -adically is reduced to writing1 -adically. Since 3 = 2 2 3 , we have1 = 2 + ( 2) 2 . The problem is then reduced to writing 2 -adically. Substituting 3 2 2 for 3 in 2 = 1 + + (3), we have 2 = 1 + + 3 2 2 = 1 + + 2 ( 2). Hence, 2 = 1 + 1 2 = 1 1 = 1 + + 2 +:::: 3 It can be seen that the -adic expansion of 2 is periodic. Thus, the -adic expansions of both1 and 3 are periodic as well. Now consider writing 1 3 - adically. Since 3 = 2 (2), we have 1 3 = 1 2 (2) . Substituting 1 1 for 2, we have 1 3 = 1 2 ( 1) = 1 2 (1 + ( 2)) = 1 2 2 + + 2 +::: : That is, the -adic expansion of 1 3 is periodic. We have found by the above ad hoc method that the-adic expansions of 3 and 1 3 are both periodic. It would be nice to find out by a principled and systematic method whether every algebraic number in Q () has a finite or periodic expansion with respect to every prime ideal in Q (), and whether it is the case for algebraic numbers in general. LetK be a number field andO its ring of integers. Let p be a prime ideal inO. To study the periodic property, we first recall the definition of the p-adic valuation p () fromO to N: p () = i () p i k O. That is, the p-adic valuation of is i if and only if i is the largest natural number such that p i divides O (i tends to infinity when = 0). The definition can be extended to algebraic numbers. We can define p () fromK toZ: p (=) = p () p () where ;2O. Consider a number fieldK and a completely splitting prime p inK. LetO be the ring of integers ofK and pO the prime ideal lying above p. Since p splits 4 completely, we have p (p) = 1 andO=pO'Z=pZ. As a result, any 2K can be written as a p-adic expansion of the form = 1 X i=k r i p i ; where k2Z and r i 2R =f0; 1;:::;p 1g for all i. However not every element of K has a finite or periodic expansion with respect to p and R. In fact it is easy to see that an element that has a finite or periodic expansion with respect to p andR must be a rational number. It is not clear that whether there is a choice of where p () = 1 and/or a choice of a representative set such that the p-adic expansions in terms of and are finite or periodic. In light of the discussion above, We define a more general notion of p-adic expansion of an algebraic number as follows. LetK be a number field andO its ring of integers. Let p be a prime ideal in O and p () the discrete valuation defined by p. Let O be a complete set of representatives of the residue fieldO=pO. A p-adic expansion of 2K with p () 0 with respect to is an expansion of the form = X i0 i ! i ; where i 2 and p (! i ) =i for all i. 5 Let i = P i j=0 j ! j . Then i modp i+1 since ! i O p = p i O p whereO p = (Op) 1 O is the local ring at p, hence, = ( i ) i0 2 lim i O=p i O = ^ O p ; where ^ O p is the local completion ofO at p. That is, as i tends to infinity, the con- vergent sequence ( i ) represents as an element in ^ O p . Conversely every element in ^ O p can be written in p-adic expansion. LetK p denote the local completion ofK atp. The definition ofp-adic expansion can be extended toK p . A p-adic expansion of 2K p with p () =d < 0 with respect to is an expansion of the form = 1 ! 0 d X i0 i ! i ; where p (! 0 d ) =d, i 2 and p (! i ) =i for all i. A p-adic expansion is finite if there isn such that i = 0 for allin. A p-adic expansion is periodic if there are i and such that for all j with iji + 1, and all k 0, j+k = j and ! j+k =! j ! k for some ! where p (!) =. Let2O where p () = 1. Then canserveasauniformizerofthecompletion ofK at p. A p-adic expansion is a -adic expansion if ! i = i for all i. A -adic expansion is periodic if there are i and such that for all j with iji + 1, and all k 0, j+k = j . 6 Let i = i1 ! i : Then a p-adic expansion of is finite when there is i = 0 for somei2N; a p-adic expansion of is periodic when there are i = j 6= 0 for some i < j. When periodic, the p-adic expansion can be placed in the form = i1 + j1 i1 1$ ji ; where$ ji =! j ! 1 i , and hence p ($ ji ) =ji. From this we observe that those 2K p with a finite or periodic p-adic expansion must be inK. The sequence ( i ) is called the dividend sequence since by substitution we can write i = i+1 i+1 + i for all i 1 where i 2 and each i+1 = ! i+1 ! 1 i is a uniformizer ofK p . We show that it is possible to control the iterative process so that it terminates with a finite or periodic p-adic expansion. More precisely, we have the following: Theorem 1.1. LetK be a number field. Let p be a prime ideal inK. LetK p be the local completion ofK at p. Then for any 2K p , there is a finite or periodic p-adic expansion of if and only if 2K. The notion of c-roundedness for algebraic numbers plays an important role in the proof of the theorem. Working with c-rounded algebraic numbers, we are able 7 to ensure the dividend sequence in thep-adic expansion is either finite or eventually periodic. When the rank of the unit group of a number field is zero, i.e. when the group of units is finite, we prove a stronger version of Theorem 1.1 as follows: Theorem 1.2. LetK be a number field that has unit rank zero. Let p be a principal prime ideal inK and a generator of p. LetK p be the local completion ofK at p. Then for any 2K p , there is a finite or periodic -adic expansion of if and only if 2K. Moreover, the time complexity of finding a -adic expansion for an element ofK is polynomial in the length of the input, the period, and p, where p is the rational prime contained in p. Theorem 1.2 applies to Q, and imaginary quadratic fields including Q (i) and Q () where is a primitive third root of unity. For a number fieldK that has unit rank greater than zero, whether there is always a finite or periodic -adic expansion for every algebraic number inK with respect to some choice of remains open. 1.2 Organization of the Thesis The thesis is organized as follows. In section 2.1, we prove an upper bound on the length of the closest vector in a lattice; in section 2.2, we apply this upper bound to the lattice spanned by the unit group of the ring of integers of a number 8 field. Section 2.3 introduces the concept of c-roundedness and proves that every algebraic number has an associate that is c-rounded using the upper bound proved in section 2.2. Theorem 1.1 is proved in chapter 3. We prove that there is always a finite or periodic p-adic expansion for every algebraic number when p is principal prime and when p is prime but not necessarily principal in section 3.1 and section 3.2, respectively. Clearly, principal prime is a special case of prime but not necessarily principal; we choose to cope with the special case first in order to tackle different issues one at a time so that the proof could be clearer to understand. Theorem 1.1 yields an algorithm for constructing p-adic expansions of elements inK where p is a prime ideal (see section 3.4). In chapter 4, we prove Theorem 1.2 and apply the associated algorithm to three specific number fields,Q (i),Q p 2 , andQ p 5 to provide some concrete numeric examples. 9 Chapter 2 c-Roundedness This chapter defines the notion of c-roundedness, which relates the absolute value of the norm with all the absolute values of the number field. We utilize c-roundedness to derive norm related bounds when proving our main theorem. We deduce an upper bound on the length of the closest vector of a lattice and then apply the upper bound to the lattice spanned by the unit group of a number field. Using this bound we prove for every algebraic number the existence of a corresponding c-rounded algebraic number. 2.1 Upper Bounding the Closest Vector In this section we examine the closest vector problem of a latticeL, and deduce an upper bound on the length of the closest vector. We denote the rank ofL as rk (L), the determinant ofL as det (L), and the length of the shortest non-zero 10 vector in the lattice as (L). We refer to [Lov, Len] for more details on lattices and to [NV] for the LLL lattice basis reduction algorithm and related applications. A full lattice inR n (whereR is the set of real numbers) is any set of the form L =L (b 1 ;:::;b n ) = ( n X i=1 i b i : i 2Z;i = 1;:::;n ) wherefb 1 ;:::;b n g is a basis ofR n . The closest vector problem is as follows: given n linearly independent vectors a 1 ;:::;a n 2R n , and a vector b2R n , find a vector v2L (a 1 ;:::;a n ) withkbvk minimal. The closest vector problem along with the shortest vector problem arise in many fields of computational mathematics and computer science [HPS, Ngu, BGJ]. To derive an upper bound on the length of the closest vector, we adapt the procedure of approximating the closest vector using Lenstra-Lenstra-Lovász lattice basis reduction algorithm (LLL algorithm) [LLL, Bab]. First, we recall that the result of Gram-Schmidt orthogonalization of an ordered basis (b 1 ;:::;b n ) of R n is another ordered basis (b 1 ;:::;b n ) whose vectorsb i ’s are pairwise orthogonal, and each vectorb j can be expressed as a linear combination of the vectors b 1 ;:::;b j as follows: b j = j X i=1 ji b i (j = 1;:::;n); (2.1) and that in this formula jj = 1. With the definition of Gram-Schmidt orthogo- nalization, the Lovász-reduced basis is defined as follows. 11 Definition 2.2 (Lovász-Reduced Basis). LetL be a lattice, (b 1 ;:::;b n ) an ordered basis ofL, and (b 1 ;:::;b n ) its Gram-Schmidt orthogonalization; let the number ij be defined by (2.1). We say that the basis (b 1 ;:::;b n ) is a Lovász-reduced basis, if the following two conditions hold: (a)j ji j 1 2 for every 1i<jn; (b) b j+1 + j+1;j b j 2 3 4 b j 2 for j = 1;:::;n 1. We then find the lower bound on the length of the shortest vector among the Gram-Schmidt orthogonalization of a Lovász-reduced basis: Lemma 2.3. LetL be a lattice inR n . Let (b 1 ;:::;b n ) be a reduced basis ofL. Let (b 1 ;:::;b n ) be the Gram-Schmidt orthogonalization of (b 1 ;:::;b n ). Then minfkb 1 k;:::;kb n kg 2 (1n) 2 kb 1 k 2 (1n) 2 (L): Proof. By property (b) in the definition of a reduced basis (Definition 2.2), we have 3 4 b j 2 b j+1 + j+1;j b j 2 = b j+1 2 + 2 j+1;j b j 2 : By property (a), we have 2 j+1;j 1 4 , and thus b j+1 2 1 2 b j 2 (1jn 1): 12 It follows by induction that b j 2 2 ij kb i k 2 (1i<jn): Hence b j 2 1j 2 kb 1 k = 2 1j 2 kb 1 k 2 1n 2 kb 1 k 2 1n 2 (L) (1jn); which proves the lemma. For any latticeL, the LLL algorithm constructs a reduced basis in a finite number of steps. Moreover, if the initial basis vectors have rational coordinates, then the reduced basis is constructed in polynomial time. We describe the result as follows. Proposition 2.4 (Lenstra et al. (1982) [LLL]). There exists a reduced basis of the latticeL =L (a 1 ;:::;a n ) for any given linearly independent vectorsa 1 ;:::;a n 2R n . Moreover, there is a polynomial time algorithm that, for any given linearly inde- pendent vectors a 1 ;:::;a n 2Q n , finds a reduced basis of the latticeL (a 1 ;:::;a n ). To prove the upper bound on the length of the approximated closest vector found using LLL algorithm, Babai proved the following: Proposition 2.5 (Babai (1986) [Bab]). Leta 1 ;:::;a n 2R n be linearly independent vectors. Let (b 1 ;:::;b n ) be a reduced basis ofL (a 1 ;:::;a n ), and (b 1 ;:::;b n ) its 13 Gram-Schmidt orthogonalization. For any given b2R n , there exists a lattice vector v2L such that bv = n X i=1 i b i ;j i j 1 2 (i = 1;:::;n): (2.6) Moreover, there is a polynomial time algorithm that, for any given linearly indepen- dent vectors a 1 ;:::;a n 2 Q n and b2 Q n , finds a lattice vector v2L (a 1 ;:::;a n ) satisfying (2.6). With the lower bound on the length of the shortest vector among the Gram- Schmidt orthogonalization of a reduced basis and the above two Propositions, we then derive the upper bound on the length of the closest vector. Lemma 2.7. Let a 1 ;:::;a n 2R n be linearly independent vectors. Let (b 1 ;:::;b n ) be a reduced basis ofL (a 1 ;:::;a n ). Then for any given b2 R n , there exists v2 L (a 1 ;:::;a n ) such that kbvk 2 2 (n1) 2 n (det (L)) 2 4 (kb 1 k) 2(n1) : Proof. Let (b 1 ;:::;b n ) be the Gram-Schmidt orthogonalization of (b 1 ;:::;b n ). Then by Proposition 2.5, there exists v2L (a 1 ;:::;a n ) such that bv = n X i=1 i b i ;j i j 1 2 (i = 1;:::;n): 14 It follows that kbvk 2 = n X i=1 i b i 2 = n X i=1 k i b i k 2 = n X i=1 j i j 2 kb i k 2 1 4 n X i=1 kb i k 2 : (2.8) Since det (L) =jdet (b 1 ;:::;b n )j =jdet (b 1 ;:::;b n )j = n Y i=1 kb i k min 1in kb i k n1 max 1in kb i k; substituting for min 1in kb i k using the inequality in Lemma 2.3, we obtain max 1in kb i k det (L) 2 1n 2 kb 1 k n1 = 2 (n1) 2 2 det (L) (kb 1 k) n1 : Thus by (2.8), kbvk 2 1 4 n X i=1 kb i k 2 n 4 max 1in kb i k 2 = 2 (n1) 2 n (det (L)) 2 4 (kb 1 k) 2(n1) : If the initial basis vectors a 1 ;:::;a n and the given vector b all have rational coordinates, then such a vector v can be constructed in polynomial time. 15 2.2 The Lattice Spanned by the Unit Group In this section we apply the upper bound deduced in section 2.1 to the lattice spannedbytheunitgroupoftheringofintegersofanumberfield. Wereferto[Neu, Section 1.7] for more details on the group of units and Dirichlet’s Unit Theorem. LetK be a number field, andO its ring of integers. LetO denote the group of units ofO. Letf 1 ;:::; r g be the real embeddings andf r+1 ; r+1 ;:::; r+s ; r+s g be the complex embeddings. Since jN ()j =j 1 ()jj r ()jj r+1 ()j 2 j r+s ()j 2 ; we define the homomorphism Log :K !R r+s Log () = (logj 1 ()j;:::; logj r ()j; 2 logj r+1 ()j;:::; 2 logj r+s ()j): Proposition 2.9 (Dirichlet’s Unit Theorem, 1846). LetK be a number field and O its ring of integers. LetO denote the group of units ofO. Then the image L := Log (O ) is a full lattice in the hyperplane H :=f(x 1 ;:::;x r+s ) :x 1 + +x r+s = 0g; 16 where r is the number of real embeddings and s the number of conjugate pairs of complex embeddings ofK. ThereforeL has rank r +s 1. Lemma 2.10. LetK be a degree n number field andO its ring of integers. LetO denote the group of units ofO. LetL := Log (O ) and b the shortest vector among a reduced basis ofL. Let a := jN()j n where 2K andH as in Proposition 2.9. Then Log (a) is contained inH, and there exists u2O such that kLog (u) Log (a)k 2 "; where " := 2 (rk(L)1) 2 (rk(L)+1)(det(L)) 2 4(kbk) 2(rk(L)1) . Proof. For 1ir +s, since i ’s are homomorphisms, we have log i jN ()j n = log i (jN ()j) i ( n ) = logj i (jN ()j)jn logj i ()j: It follows by the definition of Log that Log (a) = Log (jN ()j)n Log (): SincejN ()j = r Y i=1 j i ()j ! r+s Y i=r+1 j i ()j 2 ! , taking logs of both sides gives logjN ()j = r X i=1 logj i ()j + 2 r+s X i=r+1 logj i ()j: 17 Since n =r + 2s, it follows that r X i=1 (logjN ()jn logj i ()j) + r+s X i=r+1 2 (logjN ()jn logj i ()j) = (r + 2s) logjN ()jn r X i=1 logj i ()j + 2 r+s X i=r+1 logj i ()j ! = 0 Thus Log (a) = Log (jN ()j)n Log () is contained inH. By Proposition 2.9, sinceL = Log (O ) has rank r + s 1, we define the following r +s homomorphisms with rank r +s 1 Log i :K !R r+s1 Log i () = Log ()with the ith embedding dropped Each of these homomorphisms has a set of linearly independent vectors of size r +s 1 and shares the same determinant. Thus, by Lemma 2.7, we have kLog (u) Log (a)k 2 UB i (r +s 1) +jlogj i (u)j logj i (a)jj 2 for i = 1;:::;r, and kLog (u) Log (a)k 2 UB i (r +s 1) + 4jlogj i (u)j logj i (a)jj 2 18 fori =r + 1;:::;r +s, where UB i (k) = 2 (k1) 2 k (det (L)) 2 4 kb 1;i k 2(k1) and b 1;i is the b 1 of the ith embedding. Let UB (r +s 1) be the UB i (r +s 1) with the smallestkb 1;i k. Letkbk = min i kb 1;i k. Adding up all these r +s inequalities, we obtain (r +s)kLog (u) Log (a)k 2 (r +s) UB (r +s 1) +kLog (u) Log (a)k 2 : Hence, kLog (u) Log (a)k 2 r +s r +s 1 UB (r +s 1) = 2 (rk(L)1) 2 (rk (L) + 1) (det (L)) 2 4 (kbk) 2(rk(L)1) which proves the Lemma. In the special case where the rank of the unit lattice is 0, we have Log (u) = Log (a) = 0. The upper bound ofkLog (u) Log (a)k is thus 0. 2.3 c-Rounded Algebraic Numbers The concept ofc-roundedness is a tool we use to monitor the dividend sequence in the iterative division steps finding p-adic expansions. Using the upper bound described in Lemma 2.10, we prove for each algebraic number the existence of a corresponding c-rounded algebraic number. 19 We first compare the iterative division steps between rational integers/numbers and algebraic numbers to motivate the definition of c-roundedness. Let a be an element inN, the set of natural numbers. Let p2N be a rational primeandR =f0; 1;:::;p 1garesidueclassrepresentativeset. Thenbyapplying division iteratively, we can write a = a 1 p +r 0 and a 1 = a 2 p +r 1 and so on where r i 2R and a i 2N for all i. Observing that a>a 1 >a 2 > 0, it follows that there are finitely many sucha i ’s. Thus, thep-adic expansion ofa2N is finite since the set of natural numbers is well-ordered. Assume without loss of generality that a = P m i=0 r i p i wherer 0 6= 0,r m 6= 0, andr i = 0 for alli>m. Now consider writing a as a p-adic expansion using the same residue class representative set R. Since a + (a) = 0, we have a = (pr 0 ) + m X i=1 (p 1r i )p i + 1 X i=m+1 (p 1)p i : Then, we can writea = a 0 1 p + (pr 0 ) and a 0 1 = a 0 2 p + (p 1r 1 ) and so on where 0>a 0 i 2Z for all i. It is clear thatjajja 0 1 jja 0 2 j 1. Thus, the p-adic expansion of a negative integer is periodic since the set of negative integers is well-ordered. Letb be an element inQ, the set of rational numbers. Writeb = k n wherek2Z, n2N and (k;n) = 1. Assume the samep andR as above. First we look at the case wherejkj<n and without loss of generality we assume thatp-n. Since (p;n) = 1, 20 we have k n r modp for some r2R. That is, knr =p` for some `2Z. Since jkj<n and 0rp 1, we have 0` = knr p kn (p 1) p = k +n p n>n: We can see from the above thatjkj<n impliesn<` 0. Rewritingknr =p` as k n = ` n p +r makes it clear that ` n is the next term in the dividend sequence of computing the p-adic expansion. Sincen<` 0, the next term in the dividend sequence (namely, the quotient of ` n ) is some m n withn<m 0. Since there are finitely many such m, the p-adic expansion of b = k n is periodic. For k n such that jkjn, it can be observed that terms in the dividend sequence first decrease until some x n such thatn < x 0, then the terms remain falling betweenn and 0. So again the p-adic expansion is finite or periodic. Aswecanseefromtheanalysisabove, thep-adicexpansionofarationalnumber is periodic since the set of rational numbers is well-ordered under the metric of absolute value and the set of integers with absolute value smaller than a certain threshold is finite. Now let us look at algebraic numbers. LetK be a number field andO its ring of integers. Let pO be a principal prime ideal and a set of representatives of the residue fieldO=pO. Let be an element inK with p () 0. Since p is principal, again by iterative divisions we can write = 1 + 0 and 1 = 2 + 1 and so on where O = p, i 2 and i 2K for all i. However, in this case the absolute values are not unique. In 21 fact absolute values of elements inK can be induced by [K :Q] many embeddings fromK toR orC. Moreover, it is possible thatj i j is greater thanj i+1 j for some absolute value, and the other way around for some other absolute value. Our strategy is to find a way that allows us to take all the embeddings into consideration. This will be useful for controlling the norm of the elements that arise in the dividend sequence. To this end, we define the concept of c-roundedness. Definition 2.11 (c-Roundedness). LetK be a number field of degree n and c 1 a real number. Then 2K is c-rounded if, for all the n embeddings fromK to R orC, 1 c jN ()j 1 n j ()jcjN ()j 1 n : That is, an algebraic number isc-rounded if each of its absolute value induced by an embedding is close to thenth square root ofjN ()j within a common factor c wheren isthe numberofembeddings. Wethenprovewiththehelpof Lemma 2.10 for each algebraic number there is u2O , the unit group, such that u is c- rounded for some c. Theorem 2.12. LetK be a number field andO its ring of integers. LetO denote the group of units ofO. Let c :=e p " where " is as in Lemma 2.10. Then for any 2K, there exists u2O such that u is c-rounded. Proof. Let a = jN()j n . Then by Lemma 2.10, there exists u2O such that kLog (u) Log (a)k 2 ": 22 It follows that for each embedding , jlogj (u)j logj (a)jj 2 kLog (u) Log (a)k 2 " ) p " log j (u)j j (a)j p " )e p " j (a)jj (u)je p " j (a)j: (2.13) Since logj (a)j = log j(jN()j)j nj()j = log jN()j 1 n j()j , substituting forj (a)j in (2.13), we obtain e p " jN ()j 1 n j ()j j (u)je p " jN ()j 1 n j ()j )e p " jN ()j 1 n j (u)je p " jN ()j 1 n ; which implies that u is c-rounded. When the rank of the unit lattice is 0, sincekLog (u) Log (a)k = 0, we can take the unit u = 1 so that each u = is 1-rounded. 23 Chapter 3 Finite or Periodic p-adic Expansions In this chapter we prove that for any element in a number fieldK, there is always a finite or periodic p-adic expansion of where p is a prime ideal inK. In fact we prove that for any 2K p , the local completion ofK at p, there is a finite or periodic p-adic expansion of if and only if 2K. We prove that there is always a finite or periodic p-adic expansion for every algebraic number when p is principal prime and whenp is prime but not necessarily principal in section 3.1 and section 3.2, respectively. In fact the foundational case treated in section 3.1 is for principal ideals m, not necessarily prime, but with sufficiently large norm, where we prove that there is always a finite or periodic m-adic expansion for every algebraic number in the ring of fractions defined by m. In reducing the general case to the foundational case we need to tackle a set of issues so as to remove the restrictions on the ideals and the algebraic numbers. We choose to deal with the case of principal prime ideals first so that the proof is easier to understand. 24 Section 3.3 deals with the other direction of our main theorem in detail. Sec- tion 3.4 gives an algorithm to find a finite or periodic p-adic expansion, which is yielded from our constructive proof. 3.1 Principal Prime p In this section we prove the existence of either a finite or a periodic p-adic expansion for every algebraic number when p is a principal prime ideal. To show the existence, we first provide a constructive proof for the existence of either a finite or a periodic m-adic expansion for every algebraic number in the ring of fractions defined by m where m is a principal ideal whose norm is sufficiently large. Wethenshowthatforeveryalgebraicnumbernotintheringoffractionsdefined by a principal prime ideal p, finding its p-adic expansion can be reduced to finding thep-adic expansion of another algebraic number in the ring of fractions defined by p, and hence the principal ideal result can be applied to show the existence when the norm of p is sufficiently large. To remove the constraint that p needs to be sufficiently large, we lift p to a higher order d and choose properly a finite set of representatives for the residue ring over p d based on a complete set of representatives for the residue field over p. We prove that either a finite or a periodic p d -adic expansion can be found and the p d -adic expansion found is in fact a p-adic expansion as well. 25 3.1.1 A Constructive Proof In this subsection we provide a constructive proof to show the existence of either a finite or a periodic m-adic expansion for every algebraic number in the ring of fractions defined by m when m is a principal ideal and its norm is sufficiently large. LetK be a number field, andO its ring of integers. LetO denote the group of units ofO. Let m be a principal ideal inO and O a complete set of representatives of the residue ringO=mO. Let be a generator of m. LetS be the set of2O such that is invertible modulo m. LetO m :=S 1 O, the ring of fractions ofO with respect toS. Consider 2 O m . Then can be written as s 1 where ;s 2 O and s is invertible modulo m. Say st 1 modm, we then can find 2 such that t modm. Equivalently, s modm. Since m is principal, we can write s = where 2O, which can be rewritten as s 1 =s 1 + . Therefore, when m = O is principle and 2O m , we can write = s 1 = s 1 + , which is essential for our construction of m-adic expansion. By analogy of form we call s 1 the dividend, s 1 the quotient, and the remainder of a “division” step. A crucial point is that the denominator of the dividend and the quotient is the same throughout the iterative division process finding an m-adic expansion. This makes it possible to adopt the strategy of our main proof in [CH]. We sketch the idea below before presenting a formal proof. 26 Suppose that to construct an m-adic expansion for any 2O m K, we find inductively i 2 such that i t i modm wherest 1 modm. We use the field norm as a metric to monitor (s 1 i ), the sequence of dividends. For jN (s 1 i )j greater than a certain threshold , assuming N (m) is large enough, we prove that u i 2O , i+1 2O, and i 2 can be found such that u i s 1 i = s 1 i+1 + i andjN ( i+1 )j is strictly smaller thanjN ( i )j. Then in finitely many steps a dividend s 1 j such thatjN (s 1 j )j is smaller than or equal tothethreshold willbefound. Toguaranteethatthenormofthedividendstrictly decreases, we apply the notion of c-roundedness by multiplying the dividend by a certain unit u i . ForjN (s 1 j )jnotgreaterthanthreshold,weshowthatjN (s 1 j+1 )jisproperly bounded. Then, ifjN (s 1 j+1 )j is larger than the threshold, since it is bounded, in another finitely many steps another dividend will be again smaller than or equal to the threshold. As the number of algebraic integers with norm not greater thanjN (s)j is finite up to units, unless the m-adic expansion is finite, two associated algebraic numbers with norm not greater than the threshold, s 1 i and s 1 j , where j > i must be found in finitely many iterations. When two associated dividends are found, the constructed m-adic expansion is then periodic. 27 The following lemma will be used to show that there exists i+1 such that jN (s 1 i+1 )j is smaller thanjN (s 1 i )j whenjN (s 1 i )j is greater than a suitable threshold . Lemma 3.1. LetK be a degree n number field andO its ring of integers. Let O denote the group of units ofO. Let m := O be a principal ideal ofO where N (m)> 2 n . LetO m be the ring of fractions defined by m. Let O be a complete set of representatives of the residue ringO=mO. Let := max 1in; 2 j i ( )j where i ’s are the embeddings fromK toR orC. Let c be as in Theorem 2.12. Then for any 2O m (i.e., = s 1 where ;s2O and s is invertible modulo m) with jN ()j> := (c) n , there exists u2O and 2O such that us 1 s 1 2 and jN ()j jN ()j > N (m) 2 n > 1. Proof. By Theorem 2.12, there existsu2O such thatu isc-rounded; that is, we have for each embedding i 1 c jN ()j 1 n j i (u)jcjN ()j 1 n : Given thatjN ()j> = (c) n , we have j i (u)j 1 c jN ()j 1 n > 1 c ((c) n ) 1 n = : 28 Since2O m can be written in the forms 1 where;s2O ands has an inverse modulo m, substituting s 1 for inj i (u)j> , we have j i (s)j<j i (u)j: Since s is invertible modulo m, there exists t2O such that st 1 modm. Let ut modm, where 2 . Then, we haveu s modm. Sincem is principal, wecanfind2O suchthatu s =. Thenforeachembedding i (1in), we have j i ()jj i ()j =j i (u s)jj i (u)j +j i ( s)j j i (u)j + j i (s)j< 2j i (u)j: Hence, given the field norm is the product of all the embeddings i (1in), we have jN ()j = n Y i=1 j i ()j< n Y i=1 2j i (u)j = 2 n jN (u)j; which proves the lemma given that N () = N (m)> 2 n and N (u) = 1. Next we give a formal proof that forjN (s 1 j )j not greater than the threshold ,jN (s 1 j+1 )j is bounded. This bound implies an upper bound of the period of our constructed m-adic expansion. 29 Lemma 3.2. LetK be a degree n number field andO its ring of integers. LetO denote the group of units ofO. Let m :=O be a principal ideal ofO andO m the ring of fractions defined by m. Let O be a complete set of representatives of the residue ringO=mO. Let c be as in Theorem 2.12, and and as in Lemma 3.1. Then for any 2O m (i.e., =s 1 where ;s2O and s is invertible modulo m) withjN ()j, there exists u2O and 2O such that us 1 s 1 2 and N s 1 1 N (m) c 2 + 1 n n . Proof. By Theorem 2.12, there existsu2O such thatu isc-rounded; that is, we have for each embedding i 1 c jN ()j 1 n j i (u)jcjN ()j 1 n : Given thatjN ()j = (c) n , we have j i (u)jcjN ()j 1 n c ((c) n ) 1 n =c 2 : Since s is invertible modulo m, we can find t2O such that st 1 modm. Let ut modm, where 2 . Then, we have u s modm. Since m is principal, we can find 2O such that u s = , which can be rewritten as us 1 =s 1 . Then for each embedding i (1in), we have i s 1 = i us 1 1 j i ()j (j i (u)j +j i ( )j) c 2 + j i ()j : 30 Given the field norm is the product of all the embeddings i (1in), it follows that N s 1 = n Y i=1 i s 1 n Y i=1 c 2 + j i ()j = 1 N (m) c 2 + 1 n n : Next we show that the number of algebraic integers with norm not greater than jN (s)j is finite up to units. Lemma 3.3. LetK be a number field andO its ring of integers. Let be a positive real number. Then the set B :=fb2O :jN (b)jg is finite up to units. Proof. ThenumberofelementsofB uptounitsisclearlyboundedbythenumber of ideals ofK with norm bounded by . Since the norm of an ideal is a natural number, thereareatmostbc+1manydifferentvalues. Eachpossiblevalueofnorm is the product of at mostbc + 1 rational primes (counting multiplicity). For each rational prime there are at most n many prime ideals lying over it inK, wheren is thedegreeofextensionofKovertherationals. Consequentlythenumberofidealsof K with the given norm is bounded by n bc+1 . Therefore, B =fb2O :jN (b)jg is finite up to units. Remark 3.4. Let := jN (s)j. Given B is finite up to units, we can keep record of a representative setB of B. Whenever we find a dividend s 1 j such thatjN (s 1 j )j , we first decompose (in the ring of algebraic integers) j and 31 then compare the decomposition with the decomposition of those representatives already added inB with the same value of field norm. If no element inB is of the same decomposition as j , we add intoB a c-rounded associate of j (c is as in Theorem 2.12). The representatives are chosen to be c-rounded in order to apply Lemma 3.2 so thatjN (s 1 j+1 )j is bounded whenjN (s 1 j )j is smaller than threshold . In the next theorem we prove by construction the existence of either a finite or a periodic m-adic expansion for every algebraic number 2O m where m is a principal ideal and its norm is sufficiently large. Theorem 3.5. LetK be a number field of degree n andO its ring of integers. Let m be a principal ideal ofO where N (m) > 2 n . LetO m be the ring of fractions defined by m. Then for any2O m , either we can find a periodic m-adic expansion or the m-adic expansion is finite. Moreover, the period is bounded byjBj n log (c +c 1 ) log N (m) log N (m)n whereB is as in Remark 3.4 and c as in Theorem 2.12 Proof. LetO denote the group of units ofO. Let O be a complete set of representatives of the residue ringO=mO. Let be a generator of m. Since2O m , we can write =s 1 where;s2O ands is invertible modulo m. We iteratively find i 2 such that i s 1 i modm to construct an m-adic expansion. This iterative division process can be carried out since m is principal and s is invertible modulo m. 32 When s 1 2 , 0 = s 1 is the finite m-adic expansion. When s 1 = s 1 1 + 0 and both 0 ;s 1 1 2 , 0 +s 1 1 is the finite m-adic expansion. Since O, the m-adic expansion is finite only if s divides . When s 1 1 62 , we start to iteratively find u i 2O , i+1 2O and i 2 , such that u i s 1 i = s 1 i+1 + i for each s 1 i = 2 . Here u i is chosen so that u i s 1 i is c-rounded. Then since a generator for m is fixed, i+1 2O and i 2 are uniquely determined. If at some point s 1 i is in , the m-adic expansion is then finite. We now examine the first case whenjN (s 1 i )j> ( as in Lemma 3.1). When jN (s 1 i )j>, by Lemma 3.1, there existu i 2O , i+1 2O and i 2 such that u i s 1 i =s 1 i+1 + i andjN ( i+1 )j<jN ( i )j wherejN ( i )j decreases at a ratio larger than N(m) 2 n > 1. Thus,jN ( i )j decreases untiljN ( j )j becomes smaller than or equal to jN (s)j for some j >i in finitely many iterations. WhenjN (s 1 i )j, by Lemma 3.2, there existsu i 2O , i+1 2O and i 2 such that u i s 1 i = s 1 i+1 + i andjN (s 1 i+1 )j 1 N(m) (c 2 + 1) n n ( as in Lemma 3.1). Thus, ifjN (s 1 i+1 )j is again greater than , given the decreasing ratio larger than N(m) 2 n > 1, in finitely many iterationsjN (s 1 j )j becomes smaller or equal to for some j > i + 1. Denote the upper bound ofjN (s 1 i+1 )j as and the decreasing ratio as . Then the number of iterations needed for some j >i + 1 andjN ( j )jjN (s)j to be found is upper bounded by log log log = n log (c +c 1 ) log N (m) log N (m)n . 33 Thus, for the following 4 reasons, if we do not find some s 1 i 2 (this results in a finite m-adic expansion), we must have j = i for some j >i: Given the same s 1 2K we yield the same s 1 2K and 2 such that s 1 =s 1 + . When jN (s 1 i )j > , jN (s 1 i )j decreases until some j > i such that jN (s 1 j )j is found in finitely many iterations (due to Lemma 3.1). WhenjN (s 1 i )j,thevalueofjN (s 1 i+1 )jisbounded(duetoLemma3.2). ThesetB :=fb2O :jN (b)jjN (s)jgisfiniteuptounits(duetoLemma3.3). Since we must have j = i for somej >i (if the expansion is not finite) before the set of u i i ’s covers all the elements inB, the representative set of the set B, the period is bounded byjBj n log (c +c 1 ) log N (m) log N (m)n . 3.1.2 Ring of Fractions Defined by p In this subsection we prove the existence of either a finite or a periodic p- adic expansion for every algebraic number given that p is a principal prime ideal whose norm is sufficiently large by first reducing the problem to the case where the algebraic number is in the ring of fractions defined by p, and then applying Theorem 3.5. 34 Theorem 3.6. LetK be a number field of degree n. Let p be a principal prime ideal ofK where N (p) > 2 n . Then for any 2K either we can find a periodic p-adic expansion or the p-adic expansion is finite. Proof. LetO denote the ring of integers ofK andO p be the ring of fractions defined by p, which is exactly the localization ofO at p. Let be a generator of p. Consider 2O with p () = i2N. Since p = O is principal, we can write = i 0 where 0 2O and p ( 0 ) = 0. Now consider 2K with p () =i2Z. Since i 2O p , the unit group ofO p , we can write = i s 1 where ;s2O and ;s62 p. As a result, finding a p-adic expansion of 2K can be reduced to finding a p-adic expansion of s 1 2O p . By Theorem 3.5, s 1 has a finite or periodic p-adic expansion given that N (p)> 2 n , and now the theorem follows. 3.1.3 Lifting p In this subsection we remove the constraint that N (p) needs to be sufficiently large by lifting p to a higher order d so that N p d is sufficiently large where p is a principal prime ideal. Applying Theorem 3.5 we know that either a finite or a periodic p d -adic expansion can be found. By a proper choice of a finite set of representatives for the residue ring over p d , we conclude that the p d -adic expansion found is in fact a p-adic expansion as well. 35 Theorem 3.7. LetK be a number field. Let p be a principal prime ideal ofK. Then for any 2K either we can find a periodic p-adic expansion or the p-adic expansion is finite. Proof. LetO denote the ring of integers ofK andO p be the ring of fractions defined by p, which is exactly the localization ofO at p. Let be a generator of p. Since N (p)> 1, there exists a smallestd2N such that N p d > 2 [K:Q] . Since p is a principal prime ideal, applying the same reasoning in the proof of Theorem 3.6, we see that every element 2K can be written in the form i with 2O p and i2Z. Hence we can reduce the problem to finding a p-adic expansion of 2O p . Moreover we observe thatO p =O p d. To prove the theorem from Theorem 3.5, we consider the principal ideal p d with the norm greater than 2 [K:Q] which can be generated by d . Let be a complete set of representatives for the classes ofO=O. Then 0 := ( d1 X i=0 a i i :a i 2 ;i = 0;:::;d 1 ) is a finite set of representatives for the classes ofO= d O. Constructing ap d -adic expansion for with respect to 0 by Theorem 3.5, either we can find a periodicp d -adic expansion or thep d -adic expansion is finite. Since the residue representatives are in the form of P d1 i=0 a i i , the p d -adic expansion found for is in fact a p-adic expansion as well. 36 Moreover, by Theorem 3.5, the period of the p d -adic expansion is therefore bounded byjBj n log(c+c 1 )d log N(p) d log N(p)n whereB is as in Remark 3.4, c as in Theo- rem 2.12 and d2 N such that N p d > 2 [K:Q] . Since each representative in 0 is of length d, the period of the p-adic expansion is then bounded by djBj n log(c+c 1 )d log N(p) d log N(p)n . As the bound is proportional tojBj, the period is larger when jN (s)jgetslargersinceB isarepresentativesetofB :=fb2O :jN (b)jjN (s)jg. 3.2 Prime but Not Necessarily Principal P In this section we show the existence of either a finite or a periodic P-adic ex- pansion for every algebraic number when P is a prime but not necessarily principal ideal. To show the existence, we first apply concepts of ideal class group theory to reduce the problem of prime but not necessarily principal ideals to the case of principal ideals. By choosing a finite set of representatives properly, we show that the expansion found is actually a P-adic expansion. Since the principal ideal case only applies to algebraic numbers in the ring of fractions defined by P, we then show that we can reduce the problem to the case where algebraic number is in the ring of fractions defined by P even when P is a prime but not necessarily principal ideal. 37 3.2.1 Reducing the Case of P In a number fieldK with ring of integersO, an equivalence relation on nonzero fractional ideals I and J can be defined by I J whenever there exists a nonzero element 2K such that I = J. The equivalence classes are called the ideal classes ofK. Denote [I] the equivalence class of the ideal I. Then the multiplication [I][J] = [IJ] is well-defined and commutative. This multiplication turns the set of fractional ideal classes into an Abelian group, the so-called ideal class group. For an ideal P in a number fieldK, since it belongs to one of the equivalence class of the ideal [P], there is h2N such that [P] h = [O]. The smallest h is the order of the class. Since [P] h = [P h ] = [O], P h is a principal ideal. (We refer to [Neu, Section 1.3 and 1.6] for more details on the finiteness of the ideal class group.) Thus, we have the following: Proposition 3.8. LetK be a number field and P an ideal ofK. There existsh2N such that P h is a principal ideal. For P-adic expansions where P is not principal, we raise P to the hth power such that P h is principal and define a finite set of representatives to represent O=P h O. Then we can reduce the problem of prime (but not necessarily principal) ideals to the case of principal ideals. 38 Theorem 3.9. LetK be a number field andO its ring of integers. Let P be a prime ideal ofO andO P the ring of fractions defined by P. Then for any 2O P either we can find a periodic P-adic expansion or the P-adic expansion is finite. Proof. By Proposition 3.8, there exists h2 N such that P h is principal. Since N (P) > 1, there exists a smallest d2 N such that N P hd > 2 [K:Q] . To prove the theorem from Theorem 3.5, we consider the principal ideal P hd with the norm greater than 2 [K:Q] . Let be a complete set of representatives for the classes of O=PO. Take !2PP 2 , i.e., P (!) = 1. Then 0 := ( hd1 X i=0 a i ! i :a i 2 ;i = 0;:::;hd 1 ) is a finite set of representatives for the classes ofO=P hd O. Since 2O P , we can construct a P hd -adic expansion for with respect to 0 by Theorem 3.5. The constructed P hd -adic expansion is then either finite or eventually periodic. Since the residue representatives are in the form of P hd1 i=0 a i ! i where P (!) = 1, the P hd -adic expansion found for is in fact a P-adic expansion as well. 3.2.2 Ring of Fractions Defined by P In this subsection we prove the existence of either a finite or a periodic P-adic expansion for every algebraic number given that P is a prime ideal by first reducing 39 theproblemtothecasewherethealgebraicnumberisintheringoffractionsdefined by P and then applying Theorem 3.9. Theorem 3.10. LetK be a number field. Let P be a prime ideal ofK. Then for any 2K either we can find a periodic P-adic expansion or the P-adic expansion is finite. Proof. By Proposition 3.8, there exists h2N such that P h is principal. Let be a generator of P h . LetO denote the ring of integers ofK. Write =s 1 where ;s2O. Since s2O, P (s)2 N. Thus, we can put P (s) in the form P (s) = ih +j where i;j2N and 0 j < h. Take !2 PP 2 , i.e., P (!) = 1. Let & := s! hj . We then have &2O and P (&) =h (i + 1). Let := & (i+1) . We then have P () = P (&) + P (i+1) = 0. For any valuation 0 other than P onK, since 0 () = 0 and &2O, we have 0 () = 0 & (i+1) = 0 (&) 0: By the fact thatO =f2K : () 0 for all valuations g, we have 2O. Let :=! hj . We then have =s 1 =& 1 = i+1 1 = (i+1) 1 : 40 Since ; 2O and P () = 0 (i.e., is invertible modulo P), 1 2O P . By Theorem 3.9, 1 has a finite or periodic P-adic expansion, and now the theorem follows. Remark 3.11. From the proof of Theorem 3.10, we can see that when the principal ideal used to construct the expansion involves only one prime ideal P, we can reduce the problem to the case where the algebraic number is in the ring of fractions defined by P. Thus, every algebraic number has either a finite or a periodic P-adic expansion. In addition, a principal prime ideal is just a special case involving only one prime ideal. 3.3 The Converse LetK be a number field andO its ring of integers. Let p be a prime ideal in O and p () the discrete valuation defined by p. Let O be a complete set of representatives of the residue fieldO=pO. LetK p be the local completion ofK at p. An element 2K p with p () =` is in the form =! 0 ` X i0 i ! i ; where p (! 0 ` ) =`, i 2 and p (! i ) =i for all i. Recall that a p-adic expansion is finite if there is n such that i = 0 for all i n. Since ! 0 ` 2K, i 2 O and ! i 2K for all i, a finite p-adic expansion 41 ! 0 ` P n1 i0 i ! i is inK. Also recall that a p-adic expansion is periodic if there are i and such that for all j with i j i + 1, and all k 0, j+k = j and ! j+k =! j ! k for some ! where p (!) =. Thus, a periodic p-adic expansion can be placed in the form ! 0 ` i1 X j0 j ! j + X k0 ! k i+1 X j=i j ! j ! =! 0 ` i1 X j0 j ! j + 1 1! k i+1 X j=i j ! j ! : Since ! 0 ` ;!2K, j 2 O and ! j 2K for all j, a periodic p-adic expansion is also inK. By Theorem 3.10 and the above observation that a finite or periodic p-adic expansion is inK, we have the following characterization. Corollary 3.12. LetK be a number field. Let p be a prime ideal inK. LetK p be the local completion ofK at p. Then for any 2K p , there is a finite or periodic p-adic expansion of if and only if 2K. 3.4 An Algorithm Our constructive proof yields an algorithm constructing an p-adic expansion where p is a prime ideal as described in Algorithm 1. Inouralgorithm,weneedtofindc-roundedassociatesinlines19,and21. Tofind ac-roundedassociateusingBabai’smethod[Bab], weneedtoworkwiththeunitsin the ring of integers and the lattice spanned by the unit groupL := Log (O ). Since 42 Algorithm 1 Constructing a p-adic Expansion Precondition: K,O,O , p h =O, , and as in Lemma 3.1, c as in Theorem 2.12, 2K 1: function Construction(;K;O;O ;; ;;c) 2: Write = ` s 1 such that ;s2O, s is invertible modulo p 3: if h = 1 and N (p) 2 [K:Q] then 4: Find smallest d2N such that p d > 2 [K:Q] 5: p p d , d , n P d1 i=0 a i i :a i 2 ;i = 0;:::;d 1 o 6: else if h> 1 then 7: Find smallest d2N such that p hd > 2 [K:Q] 8: Find !2pp 2 9: p p d , d , n P hd1 i=0 a i ! i :a i 2 ;i = 0;:::;hd 1 o 10: end if 11: if s 1 2 then return ` s 1 12: else 13: Find 0 2 such that 0 s 1 modp h 14: 1 1 ( 0 s), 0 1, i 1 15: if s 1 1 2 then return ` ( 0 +s 1 1 ) 16: else 17: repeat 18: ifjN ( i )j>jN (s)j then 19: Find u i 2O such that u i i is c-rounded 20: else 21: Find u i 2O according to description in Remark 3.4 22: end if 23: Find i 2 , such that i s 1 u i i modp h 24: i+1 1 (u i i i s), i u 1 i i1 , i i + 1 25: until i 2f 1 ;:::; i1 g or s 1 i 2 26: end if 27: if s 1 i 2 then return ` i1 X k=0 k k k +s 1 i i1 i ! 28: else 29: Find j <i such that j = i . 30: 1 j1 i1 31: return ` j1 X k=0 k k k + 1 1 ij i1 X k=j k k k ! 32: end if 33: end if 34: end function 43 the Log homomorphism takes its values in R r+s and we can not really compute with real numbers but only with rationals, we refer to [Thi] on how to represent the units so that unit lattice can be represented by itsq-approximation whereq2N. A rational numberz 0 is called aq-approximation to a real numberz ifjzz 0 j< 2 q1 . Also refer to [Thi, Definitions 4.1.2-4.1.7] for definitions of q-approximations to vectors, matrices, and so on. LetK := Q () be represented by a monic irreducible polynomial f 2 Z[x] where f () = 0. In polynomial time, this representation can be transformed into description of the multiplication inK on a Q-basis ofK. The description can be encoded as a multiplication table of the basis , denoted MT ( ). With MT ( ), using algorithm FUNDAMENTAL [Thi, Algorihtm 7.2.14], a system of fundamental units can be represented. Then, Log (O ) can be represented with its q-approximation. To ensure the correctness of our algorithm, we must have enough precision to find a correct u i in lines 19, and 21 of Algorithm 1. Using the tools provided by Thiel’s dissertation [Thi], given " (" as in Lemma 2.10), we are able to determine the precision q so that we can find the coordinates of v = Log (v i ) onL such that kbvk<" by finding the coordinates of its approximation v 0 onL 0 whereL 0 is a q-approximation ofL and b 0 is a q-approximation of b = Log (a i ). 44 We need to compute the p-adic valuation of in line 2 and find inO the prime decomposition of natural numbers when computing u i in line 21. For algorithms performing these operations, We refer to [Coh, Algorithms 4.8.17 and 6.2.9]. Algorithm 1 applies to both principal and non-principal prime ideals. For a non-principal prime ideal P, we need to find the order of [P], h, as one of the inputs of Algorithm 1. We refer to [Buc, BW, Coh] for algorithms computing the order of [P]. 45 Chapter 4 Examples In this chapter, we apply the theorem to number fields of unit rank zero and the associated algorithm to three specific number fields,Q(i),Q( p 2), andQ( p 5) to provide some concrete numeric examples. 4.1 Number Fields of Unit Rank Zero A number fieldK that has unit rank zero is a number field whose group of units is finite. This happens whenK either has one real embedding or a conjugate pair of complex embeddings. In either case there is a unique absolute value determined by the embedding(s). It follows that every algebraic number inK is 1-rounded. This means that when applying the same proof strategy proving Theorem 3.5, there is no need to multiply the dividend in an iterative step by a unit to make it rounded. Consequently the resulting p-adic expansion is a -adic expansion. 46 Number fields that have unit rank zero are Q and imaginary quadratic fields. LetK =Q( p D) with D < 0 andO its ring of integers. Consider 2K. We can write = i s 1 where ;s2O and p () = p (s) = 0 using constantly many integer multiplications and integer division. For each division step, at most N () many divisions between two elements inO are required. Here N () is polynomial in p, therationalprimecontainedinp. Considertwoelementsa+b p D andc+d p D in O. The division ofa+b p D overc+d p D requires at most 6 integer multiplications and 2 integer divisions. The time complexity of each of these arithmetic operation is polynomial in the length of the input. Since there are exactly the period many division steps, the time complexity of finding a -adic expansion for 2 K is polynomial in the length of input, the period, and p. We have the following: Theorem 4.1. LetK be a number field that has unit rank zero. Let p be a principal prime ideal inK and a generator of p. LetK p be the local completion ofK at p. Then for any 2K p , there is a finite or periodic -adic expansion of if and only if 2K. Moreover, the time complexity of finding a -adic expansion for an element inK is polynomial in the length of the input, the period, and p, where p is the rational prime contained in p. 47 4.2 Gaussian Rationals LetK =Q (i) =fa +bija;b2Qg, the set of Gaussian rationals. The Gaussian integersZ[i] form the ring of integers ofQ (i). With a toy implementation in Sage [S + ] of our algorithm for quadratic number fields, we then provide some concrete numeric examples for each type of prime in K =Q (i). To present the concrete numeric examples in a simpler and clearer way, we denote = 0 + k X i=1 i i + 1 1 m k+m X i=k+1 i i as = [ 0 ; 1 ;:::; k ; k+1 ;:::; k+m ]; where k+1 ;:::; k+m is the periodic part of . 4.2.1 p is ramified The only ramified prime in Q (i) is p =h1 +ii. Let =f0; 1g representing O=pO =F 2 . Here are some numeric examples. 48 2O 2 ^ O 1+i i [1] 1i [0; 1; 1; 0; 1] 2 + 7i [1; 1; 1; 0; 1; 1; 0; 1] 8 + 3i [1; 1; 0; 1; 0; 0; 1; 1; 0; 1] 31 + 53i [0; 1; 1; 1; 0; 0; 0; 0; 1; 1; 1] 15 59i [0; 1; 1; 1; 0; 0; 0; 0; 1; 0; 0; 0; 1] 89 + 99i [0; 1; 0; 1; 0; 0; 1; 1; 1; 1; 1; 0; 1; 1; 1; 0; 1] 41 67i [0; 1; 1; 1; 0; 0; 0; 1; 1; 0; 0; 0; 0; 1] 361 + 686i [1; 0; 0; 1; 1; 1; 0; 0; 1; 1; 1; 0; 0; 0; 0; 1; 1; 1; 1; 0; 1] 1 3 [1; 0; 1; 1; 0; 1; 0; 0] 711i 5 [0; 1; 1; 1; 0; 1; 0; 0; 1; 1; 1; 0] 4.2.2 p splits The splitting primes inQ (i) are those p =ha +bii witha 2 +b 2 =p,a>jbj> 0 where p 1 mod 4 is a rational prime. Letp = 5. Then p =h2 +ii orh2ii. Take p =h2 +ii. Let p =f0; 1; 2; 3; 4g representingO=pO = F 5 . Also let q = 97. Then q =h9 + 4ii orh9 4ii. Take q =h9 4ii. Let q =f0; 1;:::; 96g representingO=qO = F 97 . Here are some numeric examples for ^ O p and ^ O q . 49 2O 2 ^ O 2+i 2 ^ O 94i i [3; 2] [75; 59; 60] 1i [3; 4; 1; 2] [23; 20] 2 + 7i [3; 0; 4; 2] [42; 18; 20] 8 + 3i [1; 1; 0; 3; 2] [23; 19; 20] 31 + 53i [0; 3; 2; 1; 3; 4; 1; 2] [29; 82; 58; 60] 15 59i [3; 0; 0; 4; 4; 3; 4] [52; 56; 61; 60] 89 + 99i [3; 2; 0; 2; 3; 4; 2] [61; 13; 19; 20] 41 67i [3; 0; 1; 2; 1; 0; 3; 2] [75; 40; 62; 60] 361 + 686i [2; 3; 3; 0; 1; 4; 4; 1; 4; 1; 2] [67; 84; 28; 40] 1 2 [3; 3; 4; 0; 0; 4] [49; 88; 31; 39] 711i 3 [3; 0; 0; 3; 4; 2; 1; 4; 4; 3; 0; 1; 0; 1; 4; 3] [83; 41; 47; 79; 73] 4.2.3 p is inert The inert primes in Q (i) are those p =hpi, where p 3 mod 4 is a rational prime. Letp =h3iand =f0; 1; 2;i; 1 +i; 2 +i; 2i; 1 + 2i; 2 + 2igrepresentingO=pO = F 3 2. Here are some numeric examples. 50 2O 2 ^ O 3 i [i] 1i [1 + 2i; 2i] 2 + 7i [2 +i; 2i] 8 + 3i [1;i; 2] 31 + 53i [1 + 2i; 1 + 2i; 2i; 1 +i] 15 59i [i; 2 +i; 1 + 2i; 0; 2i] 89 + 99i [1; 0; 2 + 2i; 2; 1 +i; 2] 41 67i [1 + 2i; 1 +i; 1 +i; 1; 2 + 2i] 361 + 686i [2 + 2i; 2; 1 +i; 1 +i; 1 + 2i; 1 + 2i; 2] 1 2 [2; 1] 711i 5 [2 + 2i; 1 +i; 2 +i; 1;i; 1 + 2i] We can tell from the examples when p is inert, a +bia +bi modp. That is, the result is actually applying modulo operations to the real and to the imaginary part respectively, then adding up those two parts. 4.3 A Real Quadratic Field LetK = Q p 2 , then its ring of integersO = Z[ p 2]. Here we provide some concrete numeric examples for each type of prime inK =Q p 2 . 51 4.3.1 p is ramified The only ramified prime in Q p 2 is p = p 2 . Let =f0; 1g representing O=pO =F 2 . Here are some numeric examples. 2O 2 ^ O p 2 1 [1; 0] 1 + p 2 [1; 1] 2 + 7 p 2 [0; 1; 1; 1; 0; 1] 8 + 3 p 2 [0; 1; 0; 1; 0; 0; 1] 31 + 53 p 2 [1; 1; 1; 0; 1; 1; 1; 0; 1; 1; 0; 1] 15 59 p 2 [1; 1; 1; 0; 1; 1; 1; 0; 0; 0; 0; 0;; 0; 1] 89 + 99 p 2 [1; 1; 1; 1; 1; 0; 0; 0; 0; 0; 1; 1; 0; 1; 1; 0] 41 67 p 2 [1; 1; 1; 0; 1; 1; 0; 1; 1; 1; 0; 1; 1; 0; 1] 361 + 686 p 2 [1; 0; 1; 1; 1; 0; 0; 0; 1; 1; 0; 0; 0; 1; 1; 0; 0; 1; 1; 0; 1] 1 3 [1; 0; 1; 0; 0] 711 p 2 5 [1; 1; 1; 0; 0; 0; 1; 0; 1; 1; 0; 1] 4.3.2 p splits The splitting primes in Q p 2 are those p = a +b p 2 with a 2 2b 2 = p, a>jbj> 0 where p 1 or 7 mod 8 is a rational prime. Let p = 7. Then p = 3 + p 2 or 3 p 2 . Take p = 3 + p 2 . Let p = f0; 1; 2; 3; 4; 5; 6g representingO=pO = F 7 . Also let q = 97. Then q = 52 13 + 6 p 2 or 13 6 p 2 . Take q = 13 6 p 2 . Let q =f0; 1;:::; 96g repre- sentingO=qO =F 97 . Here are some numeric examples for ^ O p and ^ O q . 2O 2 ^ O 3+ p 2 2 ^ O 136 p 2 1 [6; 1; 2] [96; 71; 72] 1 + p 2 [5; 2] [84; 59; 60] 2 + 7 p 2 [2; 3; 5; 4] [1; 84; 59; 60] 8 + 3 p 2 [4; 6] [47; 35; 36] 31 + 53 p 2 [5; 2; 0; 0; 1; 5; 4] [65; 29; 11; 12] 15 59 p 2 [3; 5; 1; 6; 2; 1; 4; 6; 1; 2] [65; 43; 61; 60] 89 + 99 p 2 [6; 1; 6; 0; 1; 2; 1; 3; 0; 5; 4] [77; 45; 35; 36] 41 67 p 2 [6; 2; 4; 1; 1; 4; 6; 1; 2] [24; 68; 85; 84] 361 + 686 p 2 [3; 3; 6; 6; 3; 0; 3; 2; 5; 1; 1; 3; 0; 5; 4] [26; 51; 88; 46; 48] 1 2 [4; 4; 5; 6; 0; 0; 6] [49; 84; 23; 85] 711 p 2 3 [4; 2; 3; 0; 3; 1; 6; 2; 5; 3; 1; 3] [86; 3; 53; 76] 4.3.3 p is inert The inert primes inQ p 2 are those p =hpi where where p 3 or 5 mod 8 is a rational prime. Let p =h3i and =f0; 1; 2; p 2; 1 + p 2; 2 + p 2; 2 p 2; 1 + 2 p 2; 2 + 2 p 2g repre- sentingO=pO =F 3 2. Here are some numeric examples. 53 2O 2 ^ O 3 1 [2] 1 + p 2 [1 + p 2] 2 + 7 p 2 [2 + p 2; 2 p 2] 8 + 3 p 2 [1; p 2; 2] 31 + 53 p 2 [1 + 2 p 2; 1 + 2 p 2; 2 p 2; 1 + p 2] 15 59 p 2 [ p 2; 2 + p 2; 1 + 2 p 2; 0; 2 p 2] 89 + 99 p 2 [1; 0; 2 + 2 p 2; 2; 1 + p 2; 2] 41 67 p 2 [1 + 2 p 2; 1 + p 2; 1 + p 2; 1; 2 + 2 p 2] 361 + 686 p 2 [2 + 2 p 2; 2; 1 + p 2; 1 + p 2; 1 + 2 p 2; 1 + 2 p 2; 2] 1 2 [2; 1] 711 p 2 5 [2 + 2 p 2; 1 + p 2; 2 + p 2; 1; p 2; 1 + 2 p 2] Similar to the case when p is inert inK =Q (i), we can tell from the examples, a +b p 2a +b p 2 modp. 4.4 AnImaginaryQuadraticFieldwithNon-Principal Ideals LetK =Q p 5 , then its ring of integersO =Z[ p 5]. Here we provide some concrete numeric examples for each type of prime inK =Q p 5 . 54 4.4.1 p is ramified The only two ramified primes in Q p 5 areh2i = 2; 1 + p 5 2 andh5i = p 5 2 . Let P = 2; 1 + p 5 . Since P is not principal, we take P 2 =h2i. Let P 2 = 0; 1; p 5; 1 + p 5 representingO=P 2 O. Here are some numeric examples for ^ O P . 2O 2 ^ O h2;1+ p 5i p 5 [ p 5] 1 p 5 [1 + p 5; p 5] 2 + 7 p 5 [ p 5; 1 + p 5; p 5] 8 + 3 p 5 [ p 5; p 5; 0; 1] 31 + 53 p 5 [1 + p 5; 1; 1 + p 5; 1; 1 + p 5; p 5] 15 59 p 5 [1 + p 5; 1; 1 + p 5; 1; 0; 0; p 5] 89 + 99 p 5 [1 + p 5; 1 + p 5; 1; 0; 0; 1 + p 5; p 5; 1] 41 67 p 5 [1 + p 5; 1; 1 + p 5; p 5; 1 + p 5; p 5; 1; 1 + p 5] 361 + 686 p 5 [1; 1 + p 5; 1 + p 5; p 5; 1; p 5; 0; 1 + p 5; 0; 1 + p 5; 1] 1 3 [1; 1; 0] 711 p 5 5 [1 + p 5; 1; 0; 1; 1 + p 5; p 5] We can tell from the examples when P = 2; 1 + p 5 , a +b p 5 a + b p 5 modP 2 . That is, the result is actually applying modulo operations to the real and to the imaginary part respectively, then adding up those two parts. 55 Let q = p 5 and q =f0; 1; 2; 3; 4g representingO=qO = F 5 . Here are some numeric examples for ^ O q . It can be proved that when q = p 5 , the q-adic expansion for 2O is finite. 2O 2 ^ O p 5 p 5 [0; 1] 1 p 5 [1; 4; 0; 1] 2 + 7 p 5 [2; 2; 0; 4; 0; 1] 8 + 3 p 5 [2; 3; 2] 31 + 53 p 5 [1; 3; 4; 0; 2; 2] 15 59 p 5 [0; 1; 2; 2; 1; 3; 0; 1] 89 + 99 p 5 [1; 4; 3; 1; 2; 4; 1] 41 67 p 5 [4; 3; 4; 4; 4; 3; 1; 1] 361 + 686 p 5 [4; 1; 3; 3; 1; 3; 3; 0; 0; 1] 1 2 [3; 0] 711 p 5 3 [4; 3; 2; 3; 2] 4.4.2 p splits There are two kinds of splitting primes in Q p 5 . One kind of splitting primes are those p = a +b p 5 with a 2 + 5b 2 = p, a >jbj > 0 where p 1 or 9 mod 20 is a rational prime. Let p = 29. Then p = 3 + 2 p 5 or 3 2 p 5 . Take p = 3 + 2 p 5 . Let p =f0; 1; 2;:::; 28g representingO=pO =F 29 . Here are some numeric examples for ^ O p . 56 2O 2 ^ O 3+2 p 5 p 5 [13; 12] 1 p 5 [17; 11; 12] 2 + 7 p 5 [6; 15; 12] 8 + 3 p 5 [2; 13; 12] 31 + 53 p 5 [24; 26; 14; 12] 15 59 p 5 [2; 6; 8; 12] 89 + 99 p 5 [9; 13; 20; 12] 41 67 p 5 [16; 19; 4; 13; 12] 361 + 686 p 5 [2; 26; 25; 5; 28; 23; 24] 1 2 [15; 26; 9; 27] 711 p 5 3 [3; 4; 23; 0; 24] The other kind of splitting primes are thosehqi = q;a + p 5 q;a p 5 where a 2 5 modq and q 3 or 7 mod 20 is a rational prime. Let q = 3. Then Q = 3; 1 + p 5 or 3; 1 p 5 . Take Q = 3; 1 + p 5 . Since Q is not principal, we take Q 2 = 2 p 5 . Let Q 2 =f0; 1; 2;:::; 8g representing O=Q 2 O =Z=9Z. Here are some numeric examples for ^ O Q . 57 2O 2 ^ O h3;1+ p 5i p 5 [2; 8; 5; 6] 1 p 5 [8; 6] 2 + 7 p 5 [7; 6; 4; 6] 8 + 3 p 5 [7; 2; 6] 31 + 53 p 5 [2; 7; 3; 1; 7; 6] 15 59 p 5 [5; 2; 7; 3; 8; 5; 6] 89 + 99 p 5 [1; 3; 0; 8; 5; 7; 6] 41 67 p 5 [5; 5; 3; 6; 8; 5; 6] 361 + 686 p 5 [3; 5; 5; 8; 7; 1; 2] 1 2 [5; 7; 1; 8] 711 p 5 5 [6; 0; 2; 3; 4; 7; 2; 4; 0; 7; 6; 4; 5; 3; 2; 8; 0; 3; 6; 8; 5; 8; 7; 0; 5] 4.4.3 p is inert The inert primes in Q p 5 are those p =hpi, where p 11 or 13 or 17 or 19 mod 20 is a rational prime. Let p =h11i and = 0; 1; 2;:::; 10; p 5; 1 + p 5;:::; 10 + 10 p 5 be the representation ofO=pO =F 11 2. Here are some numeric examples. 58 2O 2 ^ O 1 1 p 5 [ p 5] 1 p 5 [1 + 10 p 5; 10 p 5] 2 + 7 p 5 [2 + 7 p 5] 8 + 3 p 5 [3 + 3 p 5; 10] 31 + 53 p 5 [9 + 9 p 5; 2 + 4 p 5] 15 59 p 5 [1 + 7 p 5; 4 + 5 p 5; 10 p 5] 89 + 99 p 5 [10; 2 + 9 p 5; 10] 41 67 p 5 [3 + 10 p 5; 7 + 4 p 5; 10 + 10 p 5] 361 + 686 p 5 [2 + 4 p 5; 7 p 5; 8 + 5 p 5; 10] 1 2 [6; 5] 711 p 5 3 [6; 7 + 7 p 5; 3 + 3 p 5] We can tell from the examples when p is inert, a +bia +bi modp. That is, the result is actually applying modulo operations to the real and to the imaginary part respectively, then adding up those two parts. 59 Chapter 5 Conclusion and Future Directions In this thesis we prove that for 2K p , the local completion of a number field K at a prime ideal p, there is a finite or periodic p-adic expansion of if and only if 2K. WhenK = Q orK is an imaginary quadratic field, we prove that for 2K p the local completion of a number fieldK at a principal prime ideal p =O, there is a finite or periodic -adic expansion of if and only if 2K. Since the p-adic expansion found is either periodic or finite, it provides an exact p-adic expression, which may be useful for applications involving p-adic analysis or a local completion of number fields. It will be interesting to construct efficient algorithms for finding such p-adic expansions and to explore applications of such expansions. 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Abstract (if available)
Abstract
It is well known that a p-adic number has a finite or periodic p-adic expansion if and only if it is rational. An interesting question is whether the characterization generalizes to algebraic numbers
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Creator
Chen, Hsing-Hau
(author)
Core Title
On þ-adic expansions of algebraic numbers
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Computer Science
Publication Date
07/19/2019
Defense Date
04/26/2017
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University of Southern California
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Tag
algebraic number theory,c-roundedness,local completions,number fields,OAI-PMH Harvest,periodicity,þ-adic expansions,π-adic expansions
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English
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Huang, Ming-Deh (
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), Adleman, Leonard M. (
committee member
), Friedlander, Eric M. (
committee member
), Kamienny, Sheldon (
committee member
)
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chen.hsing.hau@gmail.com,hsinghau.chen@usc.edu
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Chen, Hsing-Hau
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Tags
algebraic number theory
c-roundedness
local completions
number fields
periodicity
þ-adic expansions
π-adic expansions