On the trace of the ring of integers of an abelian number field
Volume 62 / 1992
Acta Arithmetica 62 (1992), 383-389
DOI: 10.4064/aa-62-4-383-389
Abstract
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).
