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On extensions with Galois group \(\tilde A_ 4\). (Sur les extensions de groupe de Galois \(\tilde A_ 4\).) (French) Zbl 0784.11051

Let \(\widetilde {A}_ 4\) be the non split extension of the alternating group \(A_ 4\). Let \(K/\mathbb{Q}\) be a quartic field such that the Galois group of its normal closure \(N/\mathbb{Q}\) is \(A_ 4\). The paper deals with the associate Galois embedding problem, i.e. to find Galois extensions \(\widetilde {N}/\mathbb{Q}\) containing \(N/\mathbb{Q}\) with Galois group \(\widetilde {A}_ 4\). Let \(\widetilde {K}/\mathbb{Q}\) be an extension of degree 8, containing \(K\) such that its Galois closure is \(\widetilde {N}\). The authors give arithmetical conditions to determine fields \(\widetilde {K}\) without increasing the set of ramified primes. They compute polynomials of degre 8 with Galois group \(\widetilde {A}_ 4\), whose Galois closures solve the Galois embedding problem attached to fields defined by quartic polynomials of \(A_ 4\)-type given in [J. Buchmann, M. Pohst and J. von Schmettow, Math. Comput. 53, 387-397 (1989; Zbl 0714.11002)]. In particular, octic fields, totally real and totally complex, with minimal discriminant and Galois group \(\widetilde {A}_ 4\) over \(\mathbb{Q}\), are given. Other computations of octic polynomials of \(\widetilde {A}_ 4\)-type, without ramification conditions, can be found in [F.-P. Heider and P. Kolvenbach, J. Number Theory 19, 392- 411 (1984; Zbl 0566.12003)].
Reviewer: N.Vila (Barcelona)

MSC:

11R32 Galois theory
12F12 Inverse Galois theory
11R21 Other number fields
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