Inverse Galois Problem - A Simple Example: Cyclic Groups

A Simple Example: Cyclic Groups

It is possible, using classical results, to construct explicitly a polynomial whose Galois group over Q is the cyclic group Z/nZ for any positive integer n. To do this, choose a prime p such that p ≡ 1 (mod n); this is possible by Dirichlet's theorem. Let Q(μ) be the cyclotomic extension of Q generated by μ, where μ is a primitive pth root of unity; the Galois group of Q(μ)/Q is cyclic of order p − 1.

Since n divides p − 1, the Galois group has a cyclic subgroup H of order (p − 1)/n. The fundamental theorem of Galois theory implies that the corresponding fixed field

has Galois group Z/nZ over Q. By taking appropriate sums of conjugates of μ, following the construction of Gaussian periods, one can find an element α of F that generates F over Q, and compute its minimal polynomial.

This method can be extended to cover all finite abelian groups, since every such group appears in fact as a quotient of the Galois group of some cyclotomic extension of Q. (This statement should not though be confused with the Kronecker–Weber theorem, which lies significantly deeper.)