Explicit Galois Realization of Transitive Groups of Degree up to 15

Abstract

We describe methods for the construction of polynomials with certain types of Galois groups. As an application we deduce that all transitive groups G up to degree 15 occur as Galois groups of regular extensions of Q(t), and in each case compute a polynomial f ∈ Q[x] with Gal(f ) = G.

A Database for Field Extensions of the Rationals

“…We have a positive answer to problem 1 for all transitive groups up to degree 15, as shown in [21]. Problem 2 is inherently much more difficult.…”

A Polynomial with Galois Groups SL2(F16)

“…It is a computational challenge to construct polynomials with a prescribed Galois group; see [15] for methods and examples. Here, by the Galois group of a polynomial f ∈ Q[x] we mean the Galois group of a splitting field of f over Q together with its natural action on the roots of f in this splitting field.…”

A Polynomial with Galois GroupSL2(11)

“…ThenK has one nontrivial subfield K, which is of degree 12 and the Galois closure of K has Galois group PSL 2 (11). The first step of the construction is to compute this field of degree 12, which can be done using the methods in Klüners and Malle (2000). Here we use the polynomial f and get the following polynomial as output:…”