Fundamental Results and Algorithms in Dedekind Domains 1
Introduction
The easiest way to start studying number fields is to consider them per se, as absolute extensions ofQ;this is, for example, what we have done in [CohO].
In practice, number fields are often presented as relative extensions, represented as an algebra L/K over a base field K that is not limited to Q Consequently, fundamental algebraic structures, including the ring of integers ZL and its ideals, function not only as Z-modules but also as ZK-modules This ZK-module structure is significantly more complex and must be carefully maintained.
Regardless of the methods used to compute ZL, representing the result poses a significant challenge A key issue arises because Z-modules, including ZL and its ideals, are free and can be represented by Z-bases, such as through the Hermite normal form (HNF) This concept is further explored in [CohO, Chapter 2], and the theory can be easily generalized by making appropriate substitutions.
In the context of explicitly computable Euclidean domains, Z can be associated with a principal ideal domain (PID) under specific conditions However, ZK is generally not classified as a PID, which means that ZL does not necessarily function as a free module over ZK A straightforward example illustrating this concept is when K is defined as Q(√-10) and L is a subset of K.
Recent research by various authors, including Bos-Poh and Cohl, reveals that the challenges associated with Z-modules can be effectively addressed Most linear algebra algorithms applicable to Z-modules, as outlined in Chapter 2 of [CohO], can be seamlessly generalized to ZK-modules This chapter serves as an expanded exploration of these findings, building on the work presented in [Cohl].
This chapter focuses on finitely generated modules over Dedekind domains, providing a comprehensive overview of the key results related to these modules For additional insights, readers are encouraged to consult the works of Frobenius-Taylor or Boul.
Many theoretical results can often be demonstrated through alternative algorithmic methods Upon completing this chapter, especially the sections on the Hermite and Smith normal form algorithms in Dedekind domains, readers are encouraged to apply these algorithms to prove the results in the following section.
Finitely Generated Modules Over Dedekind Domains 2
I would like to thank J Martinet for his help in writing this section For the sake of completeness, we first recall the following definitions.
Definition 1.2.1 Let R be a domain, in other words a nonzero, commuta- tive ring with unit, and no zero divisors.
(1) We say that R is Noetherian if every ascending chain of ideals of R is finite or, equivalently, if every ideal of R is finitely generated.
(2) We say that R is integrally closed if any x belonging to the ring of frac- tions of R which is a root of amonic polynomial in R[X] belongs in fact to R.
(3) We say that R is aDedekind domain if it is Noetherian, integrally closed, and if every nonzero prime ideal of R is a maximal ideal.
Definition 1.2.2 Let R be an integral domain and K its field of fractions.
A fractional ideal is defined as a finitely generated, nonzero sub-R-module of K, or alternatively, as an R-module represented as L[d for a nonzero ideal I of R and a nonzero element d in R When d is set to 1, the fractional ideal corresponds to an ordinary ideal, which is referred to as an integral ideal.
Unless explicitly mentioned otherwise, we will always assume that ideals and fractional ideals are nonzero.
We recall the following basic facts about Dedekind domains, which explain their importance.
Proposition 1.2.3 Let R be a Dedekind domain and K its field of fractions.
(1) Every fractional ideal of R is invertible and is equal in a unique way to a product of powers of prime ideals.
(2) Every fractional ideal is generated by at most two elements, and the first one can be an arbitrarily chosen nonzero element of the ideal.
(3) (Weak Approximation Theorem) Let S be a finite set of prime ideals of
R, let (ep)pEs be a set of integers, and let (Xp)pES be a set of elements of K both indexed by 5 There exists an element x E K such that for all p E 5, vp(x - xl') = ep, while for all p ¢ 5, vp(x) ~ 0, where vp(x) denotes the p-adic valuation.
(4) If K is a number field, the ring of integersZ K of K is a Dedekind domain.
In the context of number fields, we recall the following definitions and results.
Definition 1.2.4 Let I I be a map from K to the set of nonnegative real numbers.
1.2 Finitely Generated Modules Over Dedekind Domains 3
(1) We say that I I is afield norm on K if[z]= 0 ~ x = 0, Ix+yl ::; Ixl+IYI, and Ixyl= Ixllyl for allx and y inK.
(2) We say that the norm isnon-Archimedeanif we have the stronger condi- tion Ix+yl ::; max(lxl , Iyl)for allx and y in K; otherwise, we say that the norm isArchimedean.
(3) We say that the norm istrivial ifIxl= 1 for all x =I O.
(4) We say that two norms areequivalent if they define the same topology onK.
Theorem 1.2.5 (Ostrowsky) Let K be a number field and let a, be the n = Tl + 2T2 embeddings of K into C ordered in the usual way.
(1) Let p be a prime ideal of K Set
Ixlp=N(p)-vp(:t j if x =I 0, and [O], = 0 otherwise Then Ixlp is a non-Archimedean field norm.
(2) Any nontrivial, non-Archimedean field norm is equivalent to Ixlp for a unique prime idealp.
(3) If(1 is an embedding of K into C and if we set
Ixl"= 1(1(x)1 , where I Iis the usual absolute value onC, then Ixl" is an Archimedean field norm.
(4) Any Archimedean field norm is equivalent to Ixl" for a unique a, with
1 ::;i ::;Tl + T2ã (Note that Ixl"o+r2 is equivalent to Ixl", for Tl < i ::;
Definition 1.2.6 A place of a number field K is an equivalence class of nontrivial field norms Thus, thanks to the above theorem, the places of K can be identified with the prime ideals of K together with the embeddingsa, for 1 ::;i ::;Tl + T2 •
Finally, we note the importantproduct formula (see Exercise1).
Proposition 1.2.7 Let ni = 1 for 1 ::;i ::;Tl' ni = 2 if Tl