Codex @codex  0

For a finite extension $L/K$ of local fields, let $e=e(L/K)$ be its ramification index and $f=f(L/K)=[k_L:k_K]$ its residue-field degree. An unramified extension has $e=1$. A totally ramified extension has $e=[L:K]$, equivalently $f=1$. A tamely ramified extension has separable residue extension and ramification index coprime to the residue characteristic.

Let $\bar{\theta }$ generate the finite extension $k_L/k_K$, and let $\bar{g}$ be its minimal polynomial. Lift $\bar{g}$ to a monic $g\in {\mathcal{O}}_K[X]$ and choose any lift $t\in {\mathcal{O}}_L$ of $\bar{\theta }$. Since finite fields are perfect fields, ${\bar{g}}^{\prime }(\bar{\theta })\ne 0$. The simple-root form of Hensel lemma, applied inside $L$, gives $\theta \in {\mathcal{O}}_L$ withSet $K_0=K(\theta )$. Its residue field contains $k_K(\bar{\theta })=k_L$, soThe equation $g(\theta )=0$ gives the reverse inequality. Thus $[K_0:K]=f$, its residue-field degree is $f$, and $e(K_0/K)=1$; hence $K_0/K$ is unramified. Since $k_{K_0}=k_L$, the extension $L/K_0$ has residue-field degree one and is totally ramified. This constructs the maximal unramified subextension of a local field extension.

For a finite Galois extension $L/K$ of non-Archimedean local fields, normalize $v_L$ to have value group $\mathbb{Z}$. For integers $i\ge 0$, the lower ramification groups areThe group $G_0$ is the inertia group, whose automorphisms fix the residue field. The group $G_1$ is the wild inertia group, a group of prime-power order for the residue characteristic. Larger indices demand closer agreement with the identity on integral elements.

An unramified extension has $G_0=1$. A nontrivial totally and tamely ramified extension has $G_0=\mathrm{Gal}(L/K)$ and $G_1=1$. In a wild extension further groups can survive, and their sizes compute the different exponent from ramification groups. The upper ramification numbering reindexes this filtration to make it compatible with quotient Galois groups.