On the trace of the ring of integers of an abelian number field

• Autorzy: Kurt Girstmair
• Języki publikacji: EN

Abstrakty

EN

Let K, L be algebraic number fields with K ⊆ L, and $O_K$, $O_L$ their respective rings of integers. We consider the trace map
$T = T_{L/K} : L → K$ and the $O_K$-ideal $T(O_L) ⊆ O_K$. By I(L/K) we denote the group index} of $T(O_L)$ in $O_K$ (i.e., the norm of $T(O_L)$ over ℚ). It seems to be difficult to determine I(L/K) in the general case. If K and L are absolutely abelian number fields, however, we obtain a fairly explicit description of the number I(L/K). This is a consequence of our description of the Galois module structure of $T(O_L)$ (Theorem 1). The case of equal conductors $f_K = f_L$ of the fields K, L is of particular interest. Here we show that I(L/K) is a certain power of 2 (Theorems 2, 3, 4).

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1992
• Tom: 62
• Numer: 4
• Strony: 383-389

Daty

wydano

1992

otrzymano

1992-02-10

poprawiono

1992-03-25

Twórcy

autor

Kurt Girstmair

• Institut für Mathematik, Universität Innsbruck, Technikerstr. 25/7, A-6020 Innsbruck, Österreich

Bibliografia

• [1] K. Girstmair, Dirichlet convolution of cotangent numbers and relative class number formulas, Monatsh. Math. 110 (1990), 231-256.
• [2] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, New York 1982.
• [3] H. W. Leopoldt, Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. Reine Angew. Math. 201 (1959), 119-149.
• [4] G. Lettl, The ring of integers of an abelian number field, J. Reine Angew. Math. 404 (1990), 162-170.