Finite Abelian Extension Research Articles

Extensions with Almost Maximal Depth of Ramification

The paper is devoted to classification of finite abelian extensions L/K which satisfy the condition [L:K]|D L/K. Here K is a complete discretely valued field of characteristic 0 with an arbitrary residue field of prime characteristic p, D L/K is the different of L/K. This condition means that the depth of ramification in L/K has its “almost maximal” value. The condition appeared to play an important role in the study of additive Galois modules associated with the extension L/K. The study is based on the use of the notion of independently ramified extensions, introduced by the authors. Two principal theorems are proven, describing almost maximally ramified extensions in the cases when the absolute ramification index e is (resp. is not) divisible by p-1. Bibliography: 7 titles.

Explicit units and the equivariant Tamagawa number conjecture

Let L/K be a finite abelian extension of number fields of group G . We study the Tamagawa number T Ω( L/K ) of h 0 (Spec L ), considered as a motive defined over K and with coefficients [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /][ G ]. For a large class of extensions L/K we interpret the conjectural vanishing of T Ω( L/K ) in terms of the existence of S -units in L satisfying a variety of explicit conditions. These explicit conditions are in the same spirit as, but are in general much finer than, the conditions studied by Rubin and Popescu. By using this approach we are able to complete the proof that T Ω( L/K ) vanishes for all of the genus field extensions considered by Fröhlich. In the course of proving this result we find that for certain extensions the vanishing of T Ω( L/K ) is a refinement of the main result of Solomon concerning "wild Euler systems."

On the equivariant structure of ideals in abelian extensions of local fields (with an appendix by W. Bley)

Let p be an odd rational prime and K a finite extension of ${\Bbb Q}\_p$ . We give a complete classification of those finite abelian extensions $L/K$ in which any ideal of the valuation ring of L is free over its associated order in ${\Bbb Q}\_p\[Gal(L/K)]$ . In an appendix W. Bley describes an algorithm which can be used to determine the structure of Galois stable ideals in abelian extensions of number fields. The algorithm is applied to give several new and interesting examples.

A Note on Class Field Theory for Two-Dimensional Local Rings

The class field theory for the fraction field of a two-dimensional complete normal local ring with finite residue field is established by S. Saito. In this paper, we investigate the index of the norm group in the K 2-idele class group for a finite Abelian extension of such fields and deduce that the existence theorem does not hold for almost fields in this case.

Swan Modules and Hilbert–Speiser Number Fields

A number field K is called a Hilbert–Speiser field if for each tamely ramified finite abelian extension N/K the ring of algebraic integers of N, ON, has a normal integral basis over OK, the ring of algebraic integers of K. The classical Hilbert–Speiser theorem proves that the field of rational numbers Q is such a field. It is well known that the class number of a Hilbert–Speiser field must equal 1. We consider tame elementary abelian extensions of a number field K and Swan modules to obtain additional necessary conditions for K to be a Hilbert–Speiser field. We show that among all number fields, the field Q is the only Hilbert–Speiser field.

Initial layers of Z p -extensions and Greenberg's conjecture

Let p be a prime number, K a finite abelian extension of Q containing p-th roots of unity and K n the n-th layer of the cyclotomic Z p -extension of K. Under some conditions we construct an element of K n from an ideal class of the maximal real subfield of K n . We determine whether its p-th root is contained by some Z p -extension of K n or not for each n, using the zero of p-adic L-function and the order of the ideal class group of the maximal real subfield of K m for sufficiently large m.

Base Change for Higher Stickelberger Ideals

To a finite abelian extension of algebraic number fieldsK/kwith Galois groupG, one can attach annth Stickelberger idealI(n) for each non-negative integern. Whenk′ is an intermediate field, we also consider the extensionK/k′, and show that the associatednth Stickelberger idealI′(n) is contained inI(n). Consequently, if a conjecture on the annihilation of the QuillenK-groupK2nof the ring of integersOKbyI(n) holds forK/k, then it also holds forK/k′.

Almost Completely Decomposable Groups and Integral Linear Algebra

Ž . An almost completely decomposable group X is a finite essential abelian extension of a completely decomposable group A of finite rank. Almost completely decomposable groups appeared early in studies of torsion-free abelian groups, but only as examples, usually exhibiting pathological dew x compositions. In 1974 Lady Lad74 initiated a systematic theory of such groups based on the fundamental concept of regulating subgroup. The regulating subgroups can be defined as the completely decomposable subgroups of least index in an almost completely decomposable group. Details on the subsequent developments can be found in the survey article w x Mad95 . Also see the remarks preceding Theorem 5.1. Almost completely decomposable groups are easily written down. In fact, one traditionally starts with a completely decomposable group A inside a divisible hull Q A,

Criterion for the Equality of Norm Groups of Idele Groups of Algebraic Number Fields

One of the fundamental theorems of global class field theory states that there is a one-to-one correspondence between finite Abelian extensions of an algebraic number fieldkand the norm groups of the idele class groupCk=Jk/k* ofk. More generally, for finite extensionsKandLofkthere is the following group theoretic interpretation ofNK/kCK⊆NL/kCL. LetEbe a finite Galois extension ofkcontainingKandL, and letG=G(E/k),H=G(E/K), andN=G(E/L) be the corresponding Galois groups. It follows by global class field theory thatNK/kCK⊆NL/kCLiffG′H⊆G′N, whereG′ is the commutator subgroup ofG. In the present work we prove thatNK/kJK⊆NL/kJLiff every element ofHof prime power order is conjugate inGto an element ofN. We also show that the same group theoretic condition is equivalent toN(K/k)⊆N(L/k), whereN(K/k) is the group of elements ofk* that are local norms everywhere fromKtok. We then use this group theoretic criterion to investigate the equality of norm groups as subgroups ofk*.

Über Arithmetische Assoziierte Ordnungen

Letpbe an odd rational prime, andL/Ka totally ramified finite abelian extension ofp-adic fields. LetGdenote the Galois group ofL/K. For each idealIof the valuation ring ofLwe study its “associated order”EndQp[G](I) with respect to Qp[G]. This order is a natural operating ring for the idealI, and is an important invariant of its Zp[G]-structure. We show that unlessGis cyclic no such order can contain any idempotent of Qp[G] which does not already lie in Zp[G] This is a strong restriction on the possible structures of arithmetical associated orders. We also describe an algorithm for the explicit computation of associated orders which can in many cases be implemented effectively on a computer. In this way we are able to compute explicit examples of arithmetical associated orders which falsify a conjecture previously made by the second named author.

Maximal independent systems of units in global function fields

Introduction. Around 1980, Galovich and Rosen (cf. [GR1] and [GR2]) computed the index of cyclotomic units in the full group of units in a cyclotomic function field over a rational function field over a finite field. Later, Hayes [H1] and Oukhaba [O] obtained a few index formulae of the elliptic units in some special extensions of the global function fields with some restrictions on the infinite prime. Feng and Yin [FY] constructed the maximal independent systems of cyclotomic units in finite abelian extensions over a rational function field. Recently Shu [S] extended the result in [GR1] to the global function fields with degree one infinite prime. In the Mini-Conference on the Arithmetic of Function Fields held at Brown University in April 1996, Yin announced a result along this line which extended his work in [Y] by removing the restriction on the degree of the infinite prime. This result seems to be the best one concerning the cyclotomic unit index in the sense that the base field can be any global function field. In this paper we construct maximal independent systems of units in an abelian extension over such a global function field. Notations and terminology are standard if not explained. Specifically, • k: a global function field with a constant field Fq of q elements. • ∞: a fixed infinite prime. • A: the ring of the functions in k which are holomorphic away from ∞. • M∞: the set of integral ideals of A. • e: the unit ideal of A. • P: the set of k-primes (k-places). • kv: the local field over k completed at any v ∈ P. For any m ∈M∞, • Pm := {p ∈ P : (p,m) = 1}. • Mm := {b ∈M∞ : (b,m) = 1}. • mv (resp. Av): the completion of m (resp. A) in kv.

GENUS FIELDS, CONDUCTORS AND RELATIVE INTEGRAL BASES FOR ABELIAN FUNCTION FIELDS

For a finite abelian tame extension L/Fq(T), we will determine its genus field and conductor in this paper, and prove the existence theorems of the relative integral bases for L over any of its subfields.

On the ideal class groups of imaginary abelian fields with small conductor

Let k k be any imaginary abelian field with conductor not exceeding 100, where an abelian field means a finite abelian extension over the rational field Q {\mathbf {Q}} contained in the complex field. Let C ( k ) C(k) denote the ideal class group of k k , C − ( k ) {C^ - }(k) the kernel of the norm map from C ( k ) C(k) to the ideal class group of the maximal real subfield of k k , and f ( k ) f(k) the conductor of k ; f ( k ) ⩽ 100 k;f(k) \leqslant 100 . Proving a preliminary result on 2 2 -ranks of ideal class groups of certain imaginary abelian fields, this paper determines the structure of the abelian group C − ( k ) {C^ - }(k) and, under the condition that either [ k : Q ] ⩽ 23 [k:{\mathbf {Q}}] \leqslant 23 or f ( k ) f(k) is not a prime ⩾ 71 \geqslant 71 , determines the structure of C ( k ) C(k) .

Kummer′s Criterion over Global Function Fields

Let k be a global function field over the field Fq where q = pm and let ∞ be a fixed prime of k which we assume to have degree 1. We let A be the ring of functions holomorphic outside of ∞ and we let ℘ be a prime of A. Hayes has associated to {A, ℘} a finite abelian extension k(℘) of k which is analogous to the classical cyclotomic extension Q(ζp), where ζp is a primitive pth root of unity. In this paper we define a certain set {Bi} of A-fractional ideals of k (i.e., a set of finite divisors), and we establish an analog of Kummer′s Theorem for {k(℘), Bi}.

On the Index of the Group Generated by Relative Units

Let L/ K be a finite abelian extension of algebraic number fields. We consider the group generated by "relative units" of cyclic subextensions of L/ K. We majorize its index in the full group of units of L.

On Galois isomorphisms between ideals in extensions of local fields

LetL/K be a totally ramified, finite abelian extension of local fields, let\(\mathfrak{O}_L \) and\(\mathfrak{O}\) be the valuation rings, and letG be the Galois group. We consider the powers\(\mathfrak{P}_L^r \) of the maximal ideal of\(\mathfrak{O}_L \) as modules over the group ring\(\mathfrak{O}G\). We show that, ifG has orderp m (withp the residue field characteristic), ifG is not cyclic (or ifG has orderp), and if a certain mild hypothesis on the ramification ofL/K holds, then\(\mathfrak{P}_L^r \) and\(\mathfrak{P}_L^{r'} \) are isomorphic iffr≡r′ modp m . We also give a generalisation of this result to certain extensions not ofp-power degree, and show that, in the casep=2, the hypotheses thatG is abelian and not cyclic can be removed.

Some self-dual local rings of integers not free over their associated orders

Let p be a prime number, and let K be a finite extension of the rational p-adic field ℚp. Let L/K be a finite abelian extension with Galois group G, and let L, K denote the valuation rings of L, K respectively. Then L is a free module of rank 1 over the group algebra KG. Defining the associated order of the extension L/K byL can be viewed as a module over the ring , and a fortiori over the group ring KG.

On the Galois structure of the square root of the codifferent

Let L be a finite abelian extension of ℚ, with 𝒪 L the ring of algebraic integers of L. We investigate the Galois structure of the unique fractional 𝒪 L -ideal which (if it exists) is unimodular with respect to the trace form of L/ℚ.

Norms in finite galois extensions of the rationals

We show that under certain conditions a rational number is a norm in a given finite Galois extension of the rationals if and only if this number is a local norm at a certain finite number of places in a certain finite abelian extension of the rationals.

Elliptic units and the class numbers of non-galois fields

0.1. In the field @ of complex numbers, let K be a subfield of a finite abelian extension of an imaginary quadratic field F. Our problem is to prove an explicit algebraic formula connecting the class number of K with a group of elliptic units and to give an effective algorithm of computing the class number and fundamental units of K. When K contains F, it has been solved in our previous paper [7]. We now assume that K does not contain F. Then R. Schertz [12] has derived such algebraic formulas from a result of C. Meyer [4] by ingenious matrix calculation. Embedding units by a logarithmic map into a commutative group algebra over the field [w of real numbers, we shall prove a relined formula for our algorithm in a simpler way which also makes the meaning of the matrix calculation clear. A summary of the paper has been published in [lo]. Let L be the composition of K and F, i.e., L = KF, Then the assumption above implies that L is finite abelian over F and quadratic over K. It is easy to show