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1
Content available Automatic search of rational self-equivalences
Stępień L., Stępień M. R.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2018
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Vol. 23
67--74
EN
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).
2
Content available On strong automorphisms of direct products of Witt rings. Part 2
Stępień M. R., Stępień L.
Journal of Applied Mathematics and Computational Mechanics
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2013
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Vol. 12, nr 4
119--125
EN
In the paper strong automorphisms of finitely generated Witt rings are considered. Every finitely generated Witt ring can be expressed in terms of Z / 2Z and basic indecomposable Witt rings using the operations of group ring formation and direct product. Groups of strong automorphisms of basic indecomposables and their direct products and the description of all strong automorphisms of any group Witt ring are known. In this paper the strong automorphisms of direct products of group Witt rings are considered. Presented are two wide classes of Witt rings where the group of strong automorphisms is isomorphic to the direct product of groups of strong automorphisms of Witt rings which are factors in the direct product.
3
Content available On strong automorphisms of direct products of Witt rings. Part 1
Stępień M. R., Stępień L.
Journal of Applied Mathematics and Computational Mechanics
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2013
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Vol. 12, nr 3
123--133
EN
The notion of Witt ring is fundamental in bilinear algebra. Automorphisms of Witt rings have been investigated until recent years. In this paper we consider Witt rings which are direct products of finite number of other Witt rings. We shall present a necessary condition in order to group of all strong automorphisms of direct product of Witt rings be a direct product of groups of strong automorphisms of Witt rings which are factors in the direct product. Subsequently, there are considered some examples of Witt rings, where described condition is fulfilled.
4
Content available Automatic search of automorphisms of Witt rings
Stępień L., Stępień M. R.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2011
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Vol. 16
141--146
EN
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.
5
Content available A construction of infinite set of rational self-equivalences
Stępień M.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2009
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Vol. 14
117-132
EN
In [5] it was shown that two number fields have isomorphic Witt rings of quadratic forms if and only if there is a Hilbert symbol equivalence between them. A Hilbert symbol equivalence between two number fields K and L is a pair of maps(t,T), where t: K ∗/K∗2→L∗/L∗2 is a group isomorpism and T: ΩK→Ω L is a bijection between the sets of finite and infinite primes of K and L, respectively, such that the Hilbert symbols are preserved: for any a; b∈K∗=K∗2and for any P∈ΩK,(a; b)P= (t(a), t(b))T(P) A Hilbert symbol equivalence between the field Q and itself is called rational self-equivalence. In [5] the authors present a construction of equivalence of two fields starting from the so called Hilbert small equivalence of two fields. We use this idea for constructing infinite set of rational self-equivalences.
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