Cyclic Extensions

Abstract

Continuing our discussion of binomials begun in the previous chapter, we will show that if a is a root of xn - u and if ω is a primitive n-th root of unity over F, then F(ω,α) is a splitting field for xn - u over F. Moreover, in the tower F < F (ω) < F(ω,α) the first step is a cyclotomic extension, which as we have seen, is abelian and may be cyclic. The second step is cyclic of degree d ∣ n. Nevertheless, as we will see in Chapter 13, the Galois group G F (F(ω,α)) need not even be abelian. In studying the second step in this tower, we will actually characterize finite cyclic extensions, when the base field contains appropriate roots of unity.