Galois groups and roots of solvable polynomials

In this thesis we show how to compute Galois groups of polynomials of degree 3 and 4 over fields with characteristic other than 2. We define the discriminant and show that it is a square if and only if the Galois group contains only even permutations of roots of the given polynomial. For computation of Galois groups of quartics, we define the cubic resolvent. The factorization of the resolvent together with properties of the discriminant almost characterize the Galois group of a polynomial. To distinguish the remaining groups D4 and Z4, we use the Kappe-Warren theorem. We show examples of polynomials for all Galois groups that are isomorphic to a transitive subgroup of S3 or S4.