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Journal: Annales Mathematicae Silesianae

2-ranks of class groups of Witt equivalent number fields

Kazimierz Szymiczek

Language: EN | Published: 30-09-1998 | Abstract | pp. 53-64

In [CPS] we have observed that each class of Witt equivalent quadratic number fields, except for the singleton class containing only ℚ(√-l), contains a field whose class group has 2-rank as large as we wish.
Here we generalize this observation from the case of quadratic number fields to fields of arbitrary even degree n. We prove that each class of Witt equivalent number fields of even degree n > 2 contains a field K with the 2-rank of class group as large as we wish. In fact, we prove a stronger result saying that the field in question has large 2-rank of S-class group for a finite set S of primes of K containing all infinite and all dyadic primes of the field.
We combine here an interpretation of the parity of S-class numbers in terms of a localization map (Proposition 6) with a valuation-theoretic result of Endler on the existence of fields with prescribed completions. The latter has been used in [Sz] to construct fields with prescribed Witt equivalence invariants. Here we discuss this technique again to make clear its applicability in constructing, in a given Witt class, number fields with special properties.

1998-09-30



Journal: Annales Mathematicae Silesianae

Diagonalizing the trace form in some number fields

Anna Dąbrowska

Language: EN | Published: 28-09-2001 | Abstract | pp. 17-26

In the present paper the trace form on the ring of integers of a number field is considered. All quadratic fields are determinated for which the form can be diagonalized, i.e. the quadratic fields with an integral basis orthogonal with respect to the trace. There are also given examples of fields of higher degree with the same property.

2001-09-28



Journal: Annales Mathematicae Silesianae

Fields and quadratic form schemes

Mieczysław Kula

Language: EN | Published: 30-09-1985 | Abstract | pp. 7-22

The paper presents a study of axiomatic theory of quadratic forms. Two operations on quadratic form schemes are investigated: the product of schemes and the group extension of schemes. The main result states that the product of schemes realized by fields is again realized by a field.

1985-09-30



Journal: Annales Mathematicae Silesianae

Fields and quadratic form schemes with the index of radical not exceeding 16

Lucyna Szczepanik

Language: EN | Published: 30-09-1985 | Abstract | pp. 23-46

Let g be an elementary 2-group with -1∈g and let d be a mapping of g into the family of all subgroups of g. The triple S = (g,-1,d) is called a quadratic form scheme if C1-C3 are fulfilled. The main result is: if |g| ≤ 16 or [g : R] ≤ 16 then all these schemes can be obtained as the product of schemes or schemes of form S^t from the schemes of fields C, R, F₃, F₅, Q₂(√-1), Q₂(√-2) and radical schemes S_i^β. We give a complete list of schemes for |g| ≤ 16, [g : R] ≤ 16 with all invariants.

1985-09-30



Journal: Annales Mathematicae Silesianae

An elementary proof of the d-th power reciprocity law over function fields

Anna Blaszczok

Language: EN | Published: 30-09-2011 | Abstract | pp. 49-57

This paper generalises the proof of quadratic reciprocity law in ????_q[T] presented by Chun-Gang Ji and Yan Xue [2] to the case of d-th power residues, where d divides the order of ????^*_q. Using only elementary properties of finite fields and basic number-theoretic tools we show that if P,Q∈????_q[T] are distinct irreducible polynomials then
(P/Q)_d = (-1)^q-1/ddeg(P)deg(Q)(Q/P)_d,
where (P/Q)_d is the d-th power residue symbol.

2011-09-30



Journal: Annales Mathematicae Silesianae

A classification theorem for quadratic forms over semi-local rings

Joseph L. Yucas

Language: EN | Published: 30-09-1986 | Abstract | pp. 7-12

Let R be a semi-local ring with 2∈U(R) and such that all residue class fields of R contain more than 3 elements. It is proved here that bilinear spaces over R are classified by dimension, determinant, Hasse invariant and total signature if and only if the third power of the fundamental ideal of Witt ring W(R) is torsion free. This is a generalization of the same result when R is a field due to Elman and Lam.

1986-09-30



Journal: Annales Mathematicae Silesianae

Generators of the Witt groups of algebraic integers

Alfred Czogała

Language: EN | Published: 30-09-1998 | Abstract | pp. 105-121

For a number field K let ????_K be the ring of algebraic integers of K. A basic result on the Witt ring W????_K of symmetric bilinear forms over the ring ????_K was established in [MH]. The structure of the Witt group W????_K, in terms of arithmetical invariants of K, was determined in [Sh]. Here we state precisely this description. We find generators of cyclic direct summands in the decomposition of the group W????_K into direct sum of cyclic groups. We will also describe products of these generators. This completely determines the structure of the ring W????_K. As an illustration of these results we determine the structure of Witt rings W????_K for all quadratic, and some cubic and some biquadratic fields K. The results of this paper allow us to find arithmetical conditions for the existence of an isomorphism of Witt rings W????_K → W????_L (for details see [Cz2]).

1998-09-30