book describes a constructive approach to the inverse Galois problem: Given a finite group G and a field K, determine whether there exists a Galois extension of K whose Galois group is isomorphic to G. Further, if there is such a Galois extension, find an explicit polynomial over K whose Galois group is the prescribed group G. The main theme of the book is… (More)
We describe Galois extensions where the Galois group is the quasi-dihedral, dihedral or modular group of order 16, and use this description to produce generic polynomials.
Around 1830 Galois described a procedure for assigning a finite group G to a polynomial p(x) = x n + a 1 x n−1 + · · · + a n−1 x + a n , where a 1 ,. .. , a n are rational numbers. This group (which is now called the Galois group of p(x)) " measures " the difficulty in finding the roots of this polynomial; in particular, it tells us whether or not p(x) can… (More)
We study the relationship between generic polynomials and generic extensions over a finite ground field, using dihedral extensions as an example.
Starting from a known case of generic polynomials for dihedral groups, we get a family of generic polynomials for cyclic groups of order divisible by four over suitable base fields.