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I want to prove the following statement. I'd like some feedback on part (a) and maybe a hint for part (b).

Let $K$ be a Galois extension of $F$ with $|\text{Gal}(K/F)|=20$.

(a)Prove that there exists a subfield $E$ of $K$ containing $F$ with $[E:F]=5$. Proof.

As $G=Gal(K/F)$ is a group of order $20=2^2\cdot 5$, by Sylow's theorem, there is a Sylow 2-subgroup $H$ of $G$, ie $[G:H]=5$ and so by the fundamental theorem of Galois Theory there is an intermediate field $E$ with degree 5 over $F$.

(b) Determine whether there must also exist a subfield $L$ of $K$ containing $F$ with $[L:F]=10$.

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    $\begingroup$ Can you find a group of order 20 with no subgroup of order 10? $\endgroup$ – Gerry Myerson May 28 at 0:39
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    $\begingroup$ @GerryMyerson So I think I've got it. By Cauchy's theorem there is an element of order 2 in $G$, let $H$ be the group generated by this element, thus $[G:H]=10$ so by the fundamental theorem there exists subfield $L$ such that $[L:F]=10$. Is that good? $\endgroup$ – Jimmy2Goons May 28 at 3:18
  • $\begingroup$ Looks good to me. $\endgroup$ – Gerry Myerson May 28 at 3:57
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For (b)

By Cauchy's theorem there is an element of order 2 in 𝐺, let 𝐻 be the group generated by this element, thus [𝐺:𝐻]=10 so by the fundamental theorem there exists subfield 𝐿 such that [𝐿:𝐹]=10.

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