Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 136 / 4

Krasner's lemma says that if $K$ is complete, $\alpha$ is separable over $K$, andthen $K(\alpha )\subseteq K(\beta )$.

To prove it, let $\tau$ be an automorphism of $\overline{K}$ fixing $K(\beta )$. The absolute value has a unique extension to every finite extension of the complete field $K$, so it is invariant under $\tau$. If $\tau (\alpha )\ne \alpha$, the ultrametric inequality and the displayed strict inequality giveBut $\tau (\beta )=\beta$ and invariance gives $|\tau (\alpha )-\beta |=|\alpha -\beta |$, a contradiction. Every automorphism fixing $K(\beta )$ therefore fixes $\alpha$, which proves the field inclusion by Galois correspondence.

Now let $L/{\mathbb{Q}}_p$ be finite. By the primitive element theorem, write $L={\mathbb{Q}}_p(\alpha )$ with separable minimal polynomial $f\in {\mathbb{Q}}_p[X]$. Approximate the coefficients of $f$ closely by those of a polynomial $g\in \mathbb{Q}[X]$ of the same degree. Continuity of roots over a non-Archimedean field gives a root $\beta$ of $g$ arbitrarily close to $\alpha$. Choose it close enough for Krasner's lemma; thenThe degree bound from $\mathrm{deg}g=\mathrm{deg}f$ forces equality. If $F=\mathbb{Q}(\beta )$ and $\mathfrak{p}$ is the prime selected by the embedding $F\hookrightarrow {\overline{\mathbb{Q}}}_p$, its completion isThus every finite extension of ${\mathbb{Q}}_p$ is the completion of a number field at a prime ideal.

Let $G=\mathrm{Gal}(L/K)$ and let $\mathcal{P},\mathcal{Q}$ lie over $\mathfrak{p}$. These primes define two extensions to $L$ of the $\mathfrak{p}$-adic absolute value on $K$. The conjugacy of extensions of a valuation to a normal extension says that some $\sigma \in G$ carries the first extension to the second. Equivalently, $\sigma \mathcal{P}=\mathcal{Q}$. This proves the transitivity of the Galois action on primes.

The decomposition group is the stabilizerEach such automorphism extends continuously to the completions, and restriction gives the decomposition group of a prime and local Galois group isomorphism

For the splitting field $L=\mathbb{Q}(\sqrt[3]{7},{\zeta }_3)$ of $X^3-7$, the global Galois group is $S_3$. Since $7\equiv 1(\mathrm{mod}3)$, the field ${\mathbb{Q}}_7$ contains ${\zeta }_3$. The local splitting field is therefore ${\mathbb{Q}}_7(\sqrt[3]{7})$, an Eisenstein, totally ramified cyclic extension of degree three. Hence there areprimes of $L$ above $7$, and the decomposition group of each is the normal subgroup