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Relative Galois module structure and Steinitz classes of dihedral extensions of degree 8. (English) Zbl 0953.11036

The first part of this paper deals with classes of group rings. Let \(K\) be a number field with ring of integers \({\mathcal O}_K\), and let \(G\) be a finite group. Let \(M\) be a maximal \({\mathcal O}_K\)-order in \(k[G]\) containing \({\mathcal O}_K[G]\), and let \(Cl(M)\) denote the class group of \(M\). Let \(R(M)\) denote the set of classes \(c \in Cl(M)\) such that there exist at most tamely ramified extensions \(N/K\) with Galois group \(G\) and such that the class of \(M \otimes_{{\mathcal O}_K[G]} {\mathcal O}_N\) is \(c\). If the class number of \(K\) is odd and if \(K\) does not contain a square root of \(-1\), then it is shown that \(R(M)\) coincides with the kernel of the map \(Cl(M) \rightarrow Cl(K)\) induced by augmentation \(M \rightarrow {\mathcal O}_K\), and that this kernel consists of four copies of \(Cl(K)\).
Next let \(L/K\) be an extension of number fields of relative degree \(n\), and let \({\mathcal O}_K\) and \({\mathcal O}_L\) denote their rings of integers. It is known that there is an ideal \({\mathfrak a}\) in \({\mathcal O}_K\) such that \({\mathcal O}_L \simeq ({\mathcal O}_K)^{n-1} \oplus {\mathfrak a}\). Its ideal class in the class group of \(K\) is called the Steinitz class of the extension \(L/K\).
The author proves the following results about Steinitz classes:
1) Fix a number field \(K\); for any triple \((c_1,c_2, c_3)\) of ideal classes in \(K\), there exist tame quadratic extensions \(K_i/K\) (\(i = 1, 2, 3\)) such that \(c_i\) is the Steinitz class of \(K_i/K\) and such that the compositum \(K_1K_2/K\) can be embedded in a tame octic dihedral extension.
2) The Steinitz classes of quadratic (biquadratic) extensions embeddable in tame dihedral extensions of degree 8 exhaust the class group of \(K\).
3) If \(K\) has odd class number, then the Steinitz classes of tame dihedral extensions \(N/K\) of degree 8 exhaust the class group of \(K\).

MSC:

11R65 Class groups and Picard groups of orders
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
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