We saw in Corollary 23 .17 that the cyclotomic polynomial 

s irreducible over Q for every prime p. Let ζ be a zero of ∅_p(x), and consider the field Q(ζ).
a. Show that ζ, ζ² , · · · , ζ^p-1 are distinct zeros of ∅_p(x), and conclude that they are all the zeros of ∅_p(x).
b. Deduce from Corollary 48.5 and part (a) of this exercise that G(Q(ζ)/Q) is abelian of order p - 1.
c. Show that the fixed field of G(Q(ζ)/Q) is Q. 

Data from Corollary 48.5

Let α be algebraic over a field F. Every isomorphism ψ mapping F(α) onto a subfield of F̅ such that ψ(a) = a for a ∈ F maps α onto a conjugate β of α over F. Conversely, for each conjugate β of α over F, there exists exactly one isomorphism ψ_α,β of F(α) onto a subfield of F̅ mapping α onto β and mapping each a ∈ F onto itself.

Proof Let ψ be an isomorphism of F(α) onto a subfield of F̅ such that ψ(a) = a for a ∈ F. Let irr(α, F) = a₀ + a₁x + + a_nxⁿ. Then a₀ + a₁α + · · ·a_nαⁿ = 0, so 0 ψ(a₀ + a₁α + · · ·a_nαⁿ )= a₀ +a₁ψ(α) +· · · + a_nψ(α)ⁿ, and β = ψ(α) is a conjugate of α. 

Conversely, for each conjugate β of α over F, the conjugation isomorphism ψ_α,βof Theorem 48.3 is an isomorphism with the desired properties. That ψ_α,β is the only such isomorphism follows from the fact that an isomorphism of F(α) is completely determined by its values on elements of F and its value on α.

Transcribed Image Text:

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