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On the unitary group associated to an involution of an algebraically closed field. (Sur le groupe unitaire relatif à une involution d’un corps algébriquement clos.) (French. English summary) Zbl 1261.12004

The topological group isomorphism between \(\mathbb{R}/\mathbb{Z}\) and \(\mathbb{U}=\{z\in\mathbb{C};|z|=1\}\) is well known. The author tries to extend this result to a more general situation and he proves that for any algebraically closed field \(C\) of zero characteristic and any involution \(c\) of \(C\) the groups \(U(C,c)=\{z\in C;\, zc(z)=1\}\) and \(C^{<c>}/\mathbb{Z}\) are algebraically isomorphic, where \(C^{<c>}\) is the real closed subfield of \(C\) associated with \(c\). Then he exhibits an example of an involution \(c_0\) of \(\mathbb{C}\) not conjugated in the group \(\operatorname{Aut}\mathbb{C}\) to the complex conjugacy such that \(U(\mathbb{C},c_0)\) is isomorphic as a topological group to \(\mathbb{C}^{<c_0>}/\mathbb{Z}\).

MSC:

12E30 Field arithmetic
12F10 Separable extensions, Galois theory
12J15 Ordered fields
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References:

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