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

Zbl 0569.12005
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References:

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