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Unramified extensions *October 20, 2009*

*Posted by Akhil Mathew in algebra, algebraic number theory, number theory.*

Tags: discrete valuation rings, Nakayama's lemma, unramified extensions

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Tags: discrete valuation rings, Nakayama's lemma, unramified extensions

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As is likely the case with many math bloggers, I’ve been looking quite a bit at MO and haven’t updated on some of the previous series in a while.

Back to ANT. Today, we tackle the case . We work in the local case where all our DVRs are complete, and all our residue fields are perfect (e.g. finite) (EDIT: I don’t think this works out in the non-local case). I’ll just state these assumptions at the outset. Then, **unramified extensions** can be described fairly explicitly.

So fix DVRs with quotient fields and residue fields . Recall that since , unramifiedness is equivalent to , i.e.

Now by the primitive element theorem (recall we assumed perfection of ), we can write for some . The goal is to lift to a generator of over . Well, there is a polynomial with ; we can choose irreducible and thus of degree . Lift to and to ; then of course in general, but if is the maximal ideal in , say lying over . So, we use Hensel’s lemma to find reducing to with —indeed is a unit by separability of .

I claim that . Indeed, let ; this is an -submodule of , and

because of the fact that is generated by as a field over . Now Nakayama’s lemma implies that .

Proposition 1Notation as above, if is unramified, then we can write for some with ; the irreducible monic polynomial satisfied by remains irreducible upon reduction to .

There is a converse as well:

Proposition 2If for whose monic irreducible remains irreducible upon reduction to , then is unramified, and .

Consider . I claim that . First, is a DVR. Now is a finitely generated -module, so any maximal ideal of must contain by the same Nakayama-type argument. In particular, a maximal ideal of can be obtained as an inverse image of a maximal ideal in

by right-exactness of the tensor product. But this is a field by the assumptions, so is the only maximal ideal of . This is principal so is a DVR and thus must be the integral closure , since the field of fractions of is .

Now , so unramifiedness follows.

Next up: totally ramified extensions, differents, and discriminants.

[...] Totally ramified extensions October 23, 2009 Posted by Akhil Mathew in algebra, algebraic number theory, number theory. Tags: discrete valuation rings, Eisenstein polynomials, ramification, totally ramified extensions trackback Today we consider the case of a totally ramified extension of local fields , with residue fields —recall that this means . It turns out that there is a similar characterization as for unramified extensions. [...]