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On \(p\)-groups as Galois groups. (English) Zbl 0707.12001

A group \(G\) is said to be “realizable” over a field \(K\) if there exists a Galois extension \(N/K\), the Galois group of which is isomorphic to \(G\); the field \(N\) is then called a “\(G\)-extension” of \(K\). Let \(p\) be a prime number, and denote \(D_1,D_2\) the two non-abelian groups of order \(p^3\). The paper begins with a study of the “automatic realizations” of the pair \(D_1,D_2\) in the sense of C. U. Jensen [Théorie des nombres, C. R. Conf. Int., Québec 1987, 441–458 (1989; Zbl 0696.12019)]. In the second section, the author solves the “descent problem” from a \(D_i\)-extension \((i=1,2)\) of \(K(\zeta)\) to one of \(K\), where \(K\) is any field of characteristic different from \(p\) and \(\zeta\) a primitive \(p\)-th root of unity. She proves the new and interesting result that we explain now.
Let \(L/K\) be a Galois extension with \(\Gamma = \mathrm{Gal}(L/K)\) of type \((p,p)\) and let \(A\) be a group of order \(p\). The Galois group \(\Gamma'\) of the translated extension \(L':=L(\zeta)/K':=K(\zeta)\) is canonically isomorphic to \(\Gamma\). Consider an embedding problem \((L/K,\varepsilon)\) with \(\varepsilon \in H^2(\Gamma,A)-\{0\}\) and let \(N'/K'\) be a solution of the problem \((L'/K',\varepsilon')\) where \(\varepsilon'\in H^2(\Gamma',A)\) is the translated class of \(\varepsilon\). By Kummer theory, there exists \(x\in L'^{\times}-L^{'\times p}\) such that \(N'=L'(x^{1/p})\). The key of the descent is the map \(\gamma:=n\sum_{\rho \in H} i(\rho)\rho^{-1}\) where \(\rho (\zeta)=\zeta^{i(\rho)}\) for all \(\rho \in H:= \mathrm{Gal}(K'/K)\), and \(n | H| \equiv 1 \bmod p\). Indeed, for \(N:=L'((\gamma x)^{1/p})\), we have the
Theorem: (i) \(N\) is Galois over \(K'\) with \(\mathrm{Gal}(N/K')\overset{\sim}{\longrightarrow} \mathrm{Gal}(N'/K')\); (ii) \(N\) is Galois over \(K\) with \(\mathrm{Gal}(N/K)\overset{\sim}{\longrightarrow} H\times \mathrm{Gal}(N'/K')\) (direct product).
Corollary: There exists a \(D_i\)-extension \(N_0\) of \(K\) such that \(N=N_0(\zeta)\) \((i=1,2)\).
In the last section, the author shows that there is a fixed numerical relation between the number of \(D_1\)-extensions and the number of \(D_2\)-extensions of \(K\), provided one of these numbers is finite. The proofs use the theory of groups and reviewer’s results in [J. Algebra 109, 508–535 (1987; Zbl 0625.12011)]. Note that the theorem cited above is completely explicit since there exist formulae to get all the elements \(x\) (the reviewer, loc. cit.).

MSC:

12F12 Inverse Galois theory
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