Let K be a complete discrete valuation field. (not necessarily local fields)

Definition

A  finite separable extension L/K is unramified if 

(1) e(L/K)=1.

(2) kL/kK is separable (which is always the case for local field extension, as residue fields are finite, hence perfect, and finite extension of perfect fields is separable).

Properties:

(1)

(2)

(3) Consider K subset L subset M, then M/K is unramified if and only if L/K and M/L are both unramified.  

Proposition 2.1

Let kL/kK be a separable extension, hence there exists alpha^bar in kL such that kL=kK(alpha^bar).

Let L/K be unramified, and f^bar(X) in kK[X] be the minimal polynomial of alpha^bar. Let f(X) in OK[X] be a lift of f^bar(X) with deg f^bar = deg f. Then 

(1) There exists a unique alpha in OL, such that f(alpha)=0 and alpha^bar=alpha (mod pi_L).

(2) L=K(alpha). 

Proposition 2.2

Let K be a complete discrete valuation field and l/kK be a finite separable extension. Let l=kK(alpha^bar) and f^bar(x) in kK[X] be the minimal polynomial of alpha^bar. Let f(X) in OK[X] be a lift of f^bar such that deg f =deg f^bar. Then 

(1) L=K[X]/f(X) is an unramified extension, with kL=l.

Remark:

We have 

psi:  Hom_K(L, M) --> Hom_{kK}(kL, kM).

Indeed, for any sigma in Hom_K(L, M), it induces  a map on ring of integers (notice that sigma(OL) is integral over K). This induces a map on residue field ((pi_K) subset (pi_L)).

Proposition 2.3

If L/K is unramified, then for any M, psi is a bijective.

Proposition 2.4 (Unramifiedness is preserved under compositum)

If L1/K and L2/K are unramified extensions, so is L1L2/K.

Theorem 2.5

Assume K is a local field, then for any n geq 1, there exists a unique unramified field extension of K of degree n. This field extension is a cyclic Galois extension. (the Galois group is generated by some Frobenius)