Geodesic
On the trace of the ring of integers of an abelian number field
Acta Arithmetica, Tome 62 (1992) no. 4, pp. 383-389.

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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).
DOI : 10.4064/aa-62-4-383-389

Kurt Girstmair 1

1 
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Kurt Girstmair. On the trace of the ring of integers of an abelian number field. Acta Arithmetica, Tome 62 (1992) no. 4, pp. 383-389. doi : 10.4064/aa-62-4-383-389. http://geodesic.mathdoc.fr/articles/10.4064/aa-62-4-383-389/

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