A construction of infinite set of rational self-equivalences

• Autorzy: Stępień M.
• Języki publikacji: EN

Abstrakty

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: Ω_K→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∈Ω_K,(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.

Słowa kluczowe

EN

Witt rings; Legendre symbol; Dirichlet's theorem

PL

pierścień Witta; symbol Legendre'a; twierdzenie Dirichleta

• Wydawca: Wydawnictwo Uniwersytetu Humanistyczno-Przyrodniczego im. Jana Długosza w Częstochowie
• Czasopismo: Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
• Rocznik: 2009
• Tom: Vol. 14
• Strony: 117--132
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Stępień M.

Bibliografia

• [1] J. Browkin. Teoria ciał (Fields Theory).Biblioteka Matematyczna 49,PWN, Warsaw 1978. (In Polish).
• [2] J.W.S. Cassels, A. Fröhlich (Eds.) Algebraic Number Theory. Academic Press, London 1967.
• [3] A. Czogała. On reciprocity equivalence of quadratic number fields. Acta Arithmetica ,58, No. 1, 27-46, 1981.
• [4] A. Czogała. Równoważność Hilberta ciał globalnych . (Hilbert Symbol Equivalence of Global Fields). Wydawnictwo Uniwersytetu Śląskiego, Katowice 2001. (In Polish).
• [5] R. Perlis, K. Szymiczek, P.E. Conner, R. Litherland. Matching Witts with global fields. In: W.B. Jacob, T.-Y. Lam, R.O. Robson (Eds.),Recent Advances in Real Algebraic Geometry and Quadratic Forms,(Proc. RAGSQUAD Year, Berkeley 1990-1991),Contemporary Mathematics,155, 365-387, Amer. Math. Soc. Providence, Rhode Island 1994.
• [6] J.-P. Serre. A Course in Arithmetic . Springer-Verlag, New York 1973.
• [7] K. Szymiczek. Bilinear Algebra: an Introduction to the Algebraic Theory of Quadratic Forms. Algebra, Logic and Applications Series Vol. 7, Gordon and Breach Science Publishers, Amsterdam 1997.