Finite extensions of local fields

Let $L/K$ be a finite Galois extension of nonarchimedean local fields with finite residue fields $\backslash ell/k$ and Galois group $G$. Then the following are equivalent.

• (i) $L/K$ is unramified.
• (ii) $\backslash mathcal\{O\}\_L\; /\; \backslash mathfrak\{p\}\backslash mathcal\{O\}\_L$ is a field, where $\backslash mathfrak\{p\}$ is the maximal ideal of $\backslash mathcal\{O\}\_K$.
• (iii) $[L\; :\; K]\; =\; [\backslash ell\; :\; k]$
• (iv) The inertia subgroup of $G$ is trivial.
• (v) If $\backslash pi$ is a uniformizing element of $K$, then $\backslash pi$ is also a uniformizing element of $L$.

When $L/K$ is unramified, by (iv) (or (iii)), G can be identified with $\backslash operatorname\{Gal\}(\backslash ell/k)$, which is finite cyclic.

The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K.

Again, let $L/K$ be a finite Galois extension of nonarchimedean local fields with finite residue fields $l/k$ and Galois group $G$. The following are equivalent.

• $L/K$ is totally ramified.
• $G$ coincides with its inertia subgroup.
• $L\; =\; K[\backslash pi]$ where $\backslash pi$ is a root of an Eisenstein polynomial.
• The norm $N(L/K)$ contains a uniformizer of $K$.

• Abhyankar's lemma
• Unramified morphism

{{reflist}}

• {{cite book | first=J.W.S. | last=Cassels | authorlink=J. W. S. Cassels | title=Local Fields | series=London Mathematical Society Student Texts | volume=3 | publisher=Cambridge University Press | year=1986 | isbn=0-521-31525-5 | zbl=0595.12006 |url=https://books.google.com/books?id=UY52SQnV9w4C&q=%22finite+extension%22}}
• {{cite book | last=Weiss | first=Edwin | title=Algebraic Number Theory | publisher=Chelsea Publishing | edition=2nd unaltered | year=1976 | isbn=0-8284-0293-0 | zbl=0348.12101 |url=https://books.google.com/books?id=S38pAQAAMAAJ&q=%22finite+extension%22}}

Category:Algebraic number theory