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The Galois structure of the trace form in extensions of odd prime degree. (English) Zbl 0663.12015
Let \(K/\mathbb{Q}\) denote a cyclic extension of odd prime degree \(l\) and let \(G \simeq C_l\) be its Galois group. There is a unique fractional ideal \(A\) such that \(A^2 = D^{-1}_{K/\mathbb{Q}}\), where \(D_{K/\mathbb{Q}}\) means the different. \(A\) is stable under the action of \(G\). Let \(Tr = Tr(K/\mathbb{Q})\) be the bilinear trace form of \(K/\mathbb{Q}\). On \(\mathbb{Q} G\), the group algebra, there is a form \(T\) defined by \(T(x,y) = pr_1(x\bar y)\) where for \(g \in G\), \(g = g^{-1}\) and \(pr_1 \sum (a(g)g) = a(1)\). The author proves the interesting fact that \((A,Tr)\) is \(\mathbb{Z} G\)-isometric to \((\mathbb{Z} G,T)\). He also gives a new proof in section 1 for the fact that \(A\) is \(\mathbb{Z} G\)-projective from which he deduces that \(A\) is \(\mathbb{Z} G\)-free.
Reviewer: T.Soundararajan

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R18 Cyclotomic extensions
16S34 Group rings
Full Text: DOI
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