On the ideal class groups of imaginary abelian fields with small conductor

Horie, Kuniaki; Ogura, Hiroko

doi:10.1090/S0002-9947-1995-1297529-6

On the ideal class groups of imaginary abelian fields with small conductor
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Trans. Amer. Math. Soc. 347 (1995), 2517-2532 Request permission

Abstract:

Let $k$ be any imaginary abelian field with conductor not exceeding 100, where an abelian field means a finite abelian extension over the rational field ${\mathbf {Q}}$ contained in the complex field. Let $C(k)$ denote the ideal class group of $k$, ${C^ - }(k)$ the kernel of the norm map from $C(k)$ to the ideal class group of the maximal real subfield of $k$, and $f(k)$ the conductor of $k;f(k) \leqslant 100$. Proving a preliminary result on $2$-ranks of ideal class groups of certain imaginary abelian fields, this paper determines the structure of the abelian group ${C^ - }(k)$ and, under the condition that either $[k:{\mathbf {Q}}] \leqslant 23$ or $f(k)$ is not a prime $\geqslant 71$, determines the structure of $C(k)$.

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On the relative class number of the imaginary abelian number field

9

On the relative class number of the imaginary abelian number field

10