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To resonate with KConrad's answer, the first thing that came to my mind is Serre's proof in Cassels-Fröhlich: Algebraic Number Theory (Chapter VI) that for an extension $L/K$ of local fields, the abelian part of $\mathrm{Gal}(L/K)$ is isomorphic to $K^\times/N_{L/K}L^\times$. Perhaps this theorem can be mentioned in your lecture, the statement being beautiful and simple. An important technical step in the proof is the "Ugly Lemma", in Section 1.5, about the sizes of certain group cohomology groups, and the proof starts by reducing the statement to $p$-groups using the Sylow theorems.