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Studying for my Galois theory final and came across this question (Q8.3.7) from David Cox's Galois Theory:

Let $E/F$ be a solvable Galois extension, $F$ characteristic $0$. Let $p_1,...,p_r$ be distinct primes dividing $[E:F]$. Let $\alpha = \prod_{i=1}^r p_i$.

a) Show that $F$ contains a primitive $\alpha$-th root of unity iff it contains a primitive $p_i$-th root of unity.

b) Prove that $E/F$ is radical when $F$ contains a primitive $\alpha$-th root of unity

c) Show that $E(\zeta_{\alpha})$ is a radical extension over $F$, $\zeta_{\alpha}$ a primitive $\alpha$-th root of unity.

I've done one direction of a) using Cauchy's theorem and the fact that $\zeta_m^{m/n}$ is a primitive $n$-th root of unity. But going from having a $p_i$-th primitive root of unity to an $\alpha$-th one in $F$ completely eludes me. I also don't know how to approach b) and c).

Thanks

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1 Answer 1

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A collection of hints:

  1. If $\gcd(m,n)=1$ there exist $u,v$ such that $um+vn=1$. Therefore $$\zeta_{mn}=\zeta_{mn}^{um+vn}=\zeta_n^u\cdot\zeta_m^v.$$
  2. You should use Kummer's theorem stating that if $\zeta_n\in K$ and $L/K$ is cyclic of degree $n$, then there exists $z\in L$ such that $L=K(z)$ and $z^n\in K$. I don't have Cox's book, but it sounds like this result has been covered.
  3. Show that $E(\zeta_\alpha)/F(\zeta_\alpha)$ is Galois with a solvable Galois group such that all the prime factors of the extension degree are also factors of $\alpha$.
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