Automatic search of automorphisms of Witt rings

• Autorzy: Stępień L. , Stępień M. R.
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

Witt rings; automorphisms; strong automorphisms; automorphisms of Witt rings

PL

pierścień Witta; automorfizm; automorfizm pierścieni Witta

• Wydawca: Wydawnictwo Uniwersytetu Humanistyczno-Przyrodniczego im. Jana Długosza w Częstochowie
• Czasopismo: Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
• Rocznik: 2011
• Tom: Vol. 16
• Strony: 141--146
• Opis fizyczny: Bibliogr. 4 poz., tab.

Twórcy

autor

Stępień L.

• Institute of Mathematics and Computer Science Jan Długosz University of Częstochowa al.Armii Krajowej 13/15, 42-200 Częstochowa, Poland

autor

Stępień M. R.

• Kielce University of Technology al. Tysiąclecia Państwa Polskiego 7, 25-314 Kielce, Poland

Bibliografia

• [1] C. M. Cordes. The Witt group and the equivalence of fields with respect to quadratic forms. J. Algebra, 26, 400-421, 1973.
• [2] M. Marshall. Abstract Witt Rings. Queen's Papers in Pure and Applied Math., 57, Queen's University, Ontario 1980.
• [3] R. Perlis, K. Szymiczek, P.E. Conner, R. Litherland. Matching Witts with global fields. In: Recent Advances in Real Algebraic Geometry and Quadratic Forms (Proceedings of the RAGSQUAD Year, Berkeley 1990-1991), W.B. Jacob, T.Y. Lam, R.O. Robson (Eds), Amer. Math. Soc., Contemporary Mathematics, 155, Providence, Rhode Island 1994.
• [4] M. Srebrny, L. Stępień. A propositional programming environment for linear algebra. Fundamenta Informaticae, 81, 325-345, 2007.