Recall that every finite extension L of a local field K admits a unique extension of the valuation on K, with respect to which L is complete. That makes L a local field.

Now, the discrete valuation ring of K, R, has a unique prime ideal P. Then L is called unramified if P doesn’t ramify in the integral closure of R in L; otherwise, it’s called ramified. It’s fairly straightforward to show that *n* = *ef*, where *e* is the ramification index of P (i.e. the highest power of the prime of L that divides P), and *f* is something called the inertial degree that measures the enlargement of the quotient ring of R by P. Since P is maximal, R/P is a field; if S is the integral closure of R in L and Q is the prime of S, then S/Q is a field extension of R/P of degree *f*.

As usual, unramified extensions are nice in a very precise sense. For a start, suppose that K is one of the base local fields, i.e. F((*x*)) where F = **F**(*p*), or **Q**(*p*). Then R is F[[*x*]] or **Z**(*p*) and P is (*x*) or (*p*), and the quotient ring is **F**(*p*).

I’m going to handwave another result from Galois theory, namely that for each integer *f*, there’s exactly one field extension of F of degree *f*. Further, that extension, call it **F**(*p*^*f*), has multiplicative group C(*p*^*f* – 1), so it’s generated by a primitive (*p*^*f* – 1)st root of unity. So all finite extensions of F are generated by roots of unity of order coprime to *p*.

The thing about unramified extensions of local fields is that they satisfy the exact same condition. To see why, first suppose that two extensions of K = F((*x*)) or **Q**(*p*), L and L’, are unramified and have quotient rings S/Q and S’/Q’. Suppose further that S/Q = S’/Q’, and *g* is a suitable isomorphism. If *w* generates S/Q over **F**(*p*), then *w*‘ = *g*(*w*) generates S’/Q’.

Let *h* be any monic polynomial that reduces mod P to the minimal polynomial of *w* over **F**(*p*). Obviously, in L, *h* has a root mod P, so by Hensel’s lemma, *h* has a root, say *r*. Clearly, *r* generates L over K. Similarly, L’ is generated by *r*‘, whose minimal polynomial reduces to this of *w*‘. So we can define a ring homomorphism G from L to L’ by fixing all elements of K and sending *r* to *r*‘. It’s not especially difficult to see that G is an isomorphism, so L = L’.

This theorem requires L to be unramified, because otherwise P is not prime in L, which makes Hensel’s lemma inapplicable.

Conversely, if **F**(*p*^*f*) is an extension of **F**(*p*), then let *w* generate **F**(*p*^*f*) and let *h*‘ be the minimal polynomial of *w*. If *h* reduces mod P to *h*‘, then extracting a root of *h* will yield an extension of K of degree *f*.

In slightly less technical language, the unramified extensions of **Q**(*p*) and F((*x*)) are all generated by roots of unity of order coprime to *p*.