Unramified extension of a nonarchimedean local field

Let $L/F$ be a finite extension of nonarchimedean local fields. Write

[L:F]=e(L/F)f(L/F),

where $e$ is the ramification index and $f$ the residue degree. The extension is unramified if

e(L/F)=1

and the residue extension $k_L/k_F$ is separable. For local fields with finite residue fields, separability is automatic.

Equivalently,

[L:F]=[k_L:k_F]

and a uniformizer of $F$ remains a uniformizer of $L$ .

Classification

Inside a fixed algebraic closure, for every $n\geq1$ there is a unique unramified extension of degree $n$ . It is Galois and cyclic. Reduction gives

\operatorname{Gal}(L/F) \simeq \operatorname{Gal}(k_L/k_F).

The arithmetic Frobenius acts on $k_L$ by $x\mapsto x^{q_F}$ and generates this cyclic group; geometric Frobenius is its inverse.

Maximal unramified extension

The union $F^{\mathrm{ur}}$ of all finite unramified extensions has Galois group $\widehat{\mathbb Z}$ . The Weil group retains the dense subgroup $\mathbb Z$ generated by a chosen Frobenius.

Langlands role

An unramified representation or parameter is trivial on the inertia subgroup. Its local data are therefore determined by the semisimple conjugacy class of one Frobenius element, producing the Satake parameter.

References

Jean-Pierre Serre, Local Fields, Springer, 1979.