Some polynomials over $\mathbb{Q}(t)$ and their Galois groups

Abstract

Abstract. Examples of polynomials with Galois group over Q(t) corresponding to every transitive group through degree eight are calculated, constructively demonstrating the existence of an infinity of extensions with each Galois group over Q through degree eight. The methods used, which for the most part have not appeared in print, are briefly discussed.

Some remarks on the non-real roots of polynomials

“…Galois groups over the transcendental extension Q(t) are of interest due to the Hilbert irreducibility theorem. Computing Galois groups of polynomials over Q(t) we refer to [9] and [6]. If g(x) is totally complex, (n − s) is even, and a s > 0 then f (β, x) is totally complex for all β > max{α | ∆ f,x (α) = 0}.…”

“…Galois groups over the transcendental extension Q(t) are of interest due to the Hilbert irreducibility theorem. Computing Galois groups of polynomials over Q(t) we refer to [9] and [6].…”

Constructions de polynômes génériques à groupe de Galois résoluble

“…Notons le point rationnel Q d'ordre infini sur la courbe (2.4) : [9] et [10] G. Smith donne une méthode pour déterminer un polynôme générique pour les extensions cycliques de degré 5. Après simplification on obtient…”

Computation of Galois groups over function fields

“…For each group in Tables 4 and 5 with such a system of imprimitivity, we give the associated polynomial in x 2 . Gene Smith [20] has provided test polynomials over Q(t) for all Galois groups of degree ≤ 8.…”