Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 136 / 4

Suppose first that $L=K(\alpha )$ and the degree-$n$ minimal polynomial of $\alpha$ is an Eisenstein polynomial. It is irreducible, $\alpha$ is a uniformizer of $L$, and its valuation shows that $e(L/K)\ge n=[L:K]$. Equality follows, so $L/K$ is a totally ramified extension.

Conversely, suppose $L/K$ is totally ramified of degree $n$ and choose a uniformizer $\alpha$ of $L$. The field $K(\alpha )$ already has ramification index at least $n$, so it equals $L$. Letbe the minimal polynomial. All conjugates of $\alpha$ have $L$-valuation one. Each $a_i$ with $i<n$ is an elementary symmetric polynomial in products of at least one conjugate and therefore has positive $L$-valuation. Since valuations of elements of $K$ are multiples of $n$, every $a_i$ lies in the maximal ideal of ${\mathcal{O}}_K$. Moreoverso $a_0$ is not divisible by the square of that ideal. Thus $f$ is Eisenstein, proving the Eisenstein generator of a totally ramified extension criterion.

Put ${\pi }_n={\zeta }_{p^n}-1$. The shifted cyclotomic polynomialis Eisenstein at $p$. Therefore it is irreducible, ${\pi }_n$ is a uniformizer, andso the extension is totally ramified. It is the splitting field of ${\Phi }_{p^n}$, and every automorphism is uniquelyThis proves the cyclotomic extension of a p-adic field isomorphismRestriction in the cyclotomic tower corresponds to reduction of $a$, so taking the inverse limit gives

The Local Kronecker-Weber theorem says that every finite abelian extension of ${\mathbb{Q}}_p$ is contained in a cyclotomic extension obtained by adjoining roots of unity. It identifies the totally ramified cyclotomic part throughand the maximal unramified part through its Frobenius generator.

Choose the normalization in which a uniformizer maps to arithmetic Frobenius. For $x=p^{r}u$ with $r\in \mathbb{Z}$ and $u\in {\mathbb{Z}}_p^{\times }$, define ${\mathrm{Art}}_{{\mathbb{Q}}_p}(x)$ to act by ${\mathrm{Frob}}^r$ on the maximal unramified extension and byon every $p$-power root of unity. These compatible actions define an element of the abelian Weil group, because its residue action is an integral power of Frobenius. The resulting continuous homomorphismis the Local Artin map; reversing both Frobenius conventions replaces the displayed inverse by the corresponding opposite normalization.