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Polynomials with Frobenius groups of prime degree as Galois groups. II. (English) Zbl 0598.12009

Let \(F_{p\ell}\) denote the Frobenius group of prime degree \(p\), \(p\geq 5\), and order \(p\ell\), \(\ell | p-1\). Since \(F_{p\ell}\) is a solvable group, \(F_{p\ell}\) appears as Galois group over \({\mathbb Q}\). This paper concerns the effective realization of \(F_{p\ell}\) as Galois group over \({\mathbb Q}\). First, the authors give characterization theorems for polynomials of prime degree \(p\geq 5\) over \({\mathbb Q}\) with \(F_{p\ell}\) as a Galois group over \({\mathbb Q}\), in terms of the factorization over \({\mathbb Q}\) of associated linear resolvent polynomials. Using Chebyshev polynomials of the first kind, they construct a family of polynomials over \({\mathbb Q}(u,v)\) with Galois group \(F_{p(p-1)/2}\) \((p\equiv 3 \pmod 4)\) and they give effective specializations for \(u,v\in {\mathbb Z}\). Finally, explicit examples of polynomials over \({\mathbb Q}\) with Galois group \(F_{20}\), \(F_{21}\), \(F_{55}\), and \(F_{2p}=D_ p\) \((p\leq 19)\) are given.
[For part I, cf. C. R. Math. Acad. Sci., Soc. R. Can. 7, 171–175 (1985; Zbl 0569.12005).]

MSC:

11R32 Galois theory
12F12 Inverse Galois theory
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
20F29 Representations of groups as automorphism groups of algebraic systems

Citations:

Zbl 0569.12005
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References:

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