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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

##### MSC:
 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers 11R18 Cyclotomic extensions 16S34 Group rings
##### Keywords:
cyclic extension; trace form
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##### References:
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