Primitive roots of unity in Galois Theory

Question

Why do primitive roots of unity show up and are used in proofs in the context of Galois theory? I refer for example to chapter 8.8 of Ian Stewart's "Galois theory" and to John Stillwell's paper "Galois theory for beginners".

• I quote "Theorem 2" of Stillwell's paper. If $E \supseteq B(\alpha) \supseteq B$ are fields with $\alpha^p \in B$ for some prime $p$, and if $B(\alpha)$ contains no $p$th roots of unity not in $B$ unless $\alpha$ itself is a $p$th root of unity, then $Gal(E/B(\alpha))$ is a normal subgroup of $Gal(E/B)$ and $Gal(E/B)/Gal(E/B(\alpha))$ is abelian.
• In Ian Stewart's book instead the primitive roots of unity first show up in the proof of the so-called "Natural Irrationalities" theorem. A technical lemma says that if $M$ is a subfield of $L$ containing the field $K$ and $\alpha \in M$ with $\alpha$ not a $p$th power in $M$, then the polynomial $m(x) = x^p-\alpha$ is irreducible over $M$. I can follow the logical steps of the proof but the use of primitive roots of unity is unexpected to me.

Answer 1

Hint: The roots of unity arise quite naturally in this case. Say that you have the polynomial $x^p-a$, and that it's irreducible (basically $a$ is not a $p^{th}$ root in the ground field) and separable. Let $\alpha$ be a root of $x^p-a$. What will the other roots be?

$\alpha \zeta^k$ where $\zeta$ is a primitive $p^{th}$ root of unity and $1\leq k\leq p-1$. So the splitting fields of the polynomials $x^p-a$ are connected to the $p^{th}$ roots of unity. Hence their appearance in the Galois Theory results you mention.