Wyniki wyszukiwania - BazTech

Ograniczanie wyników

3 Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

Znaleziono wyników: 3

Liczba wyników na stronie

Wyniki wyszukiwania

Wyszukiwano:
w słowach kluczowych: automorphisms of Witt rings

Sortuj według:

Ogranicz wyniki do:

Automatic search of rational self-equivalences

Stępień L., Stępień M. R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2018

Vol. 23

67--74

Two Witt rings that are not strongly isomorphic (i.e., two Witt rings over two fields that are not Witt equivalent) have different groups of strong automorphisms. Therefore, the description of a group of strong automorphisms is different for almost every Witt ring, which requires the use various tools in proofs. It is natural idea to use computers to generate strong automorphisms of the Witt rings, which is especially effective in the case of the finitely generated Witt rings, where a complete list of strong automorphisms can be created. In this paper we present the algorithm that was used to generate strong automorphisms from the infinite group of strong automorphisms of the Witt ring of rational numbers W(Q).

Automorphisms of Witt rings of finite fields

Stępień M. R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2011

Vol. 16

67--70

The problem of general description of the group of automorphisms of any Witt ring W seems to be very difficult to solve. However, there are many types of Witt rings, which automorphism are described precisely (e.g. [1], [2], [4], [5], [6],[7], [8]). In our paper we characterize automorphisms of abstract Witt rings (cf. [3]) isomorphic to powers of Witt rings of quadratic forms with coefficients in finite fields with characteristic different from 2.

Automatic search of automorphisms of Witt rings

Stępień L., Stępień M. R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2011

Vol. 16

141--146

The investigation of strong automorphisms of Witt rings is a difficult task because of variety of their structures. Cordes Theorem, known in literature as Harrison-Cordes criterion (cf. [1, Proposition 2.2], [3, Harrison's Criterion]), makes the task of describing all the strong automorphisms of a given (abstract) Witt ring W = (G, R) easier. By this theorem, it suffices to find all such automorphisms ơ of the group G that map the distiguished element -1 of the group G into itself (i.e. ơ(-1) = -1) in which the value sets of 1-fold Pfister forms are preserved in the following sense: ơ(D(1, α)) = D(1, ơ(α)) for all α ∈ G. We use the above criterion and the well-known structure of the group G as a vector space over two-element field F2 for searching all automorphisms of this group. Then we check Harrison-Cordes criterion for found automorpisms and obtain all the automorpisms of a Witt ring W. The task is easy for small rings (with small groups G). For searching of all strong automorpisms of bigger Witt rings we use a computer which automatizes the procedure described above. We present the algorithm for finding strong automorphisms of a Witt rings with finite group G and show how this algorithm can be optimized.