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

Let K, L be algebraic number fields with K ⊆ L, and ${\displaystyle O_K}$, ${\displaystyle O_L}$ their respective rings of integers. We consider the trace map ${\displaystyle T=T_{\frac{L}{K}}:L\to K}$ and the ${\displaystyle O_K}$-ideal ${\displaystyle T\left(O_L\right)\subseteq O_K}$. By I(L/K) we denote the group index} of ${\displaystyle T\left(O_L\right)}$ in ${\displaystyle O_K}$ (i.e., the norm of ${\displaystyle T\left(O_L\right)}$ 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 ${\displaystyle T\left(O_L\right)}$ (Theorem 1). The case of equal conductors ${\displaystyle f_K=f_L}$ 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).