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.)
Read more about this topic: Inverse Galois Problem
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