Cyclic Extensions

Abstract

Continuing our discussion of binomials begun in the previous chapter, we will show that if S is a splitting field for the binomial xⁿ − u, then S = F(ω, α) where ω is a primitive nth root of unity. In the tower

$$ F < F\left( \omega \right) < F\left( {\omega ,\alpha } \right) $$

the first step is a cyclotomic extension, which, as we have seen, is abelian and may be cyclic. In this chapter, we will see that the second step is cyclic of degree. d | n and α can be chosen so that min(α, F(ω)) = x^d − α^d. Nevertheless, as we will see in the next chapter, the Galois group G_F(S need not even be abelian.