Hensel’s lemma and a classification theorem September 2, 2009Posted by Akhil Mathew in algebraic number theory, algebra, number theory, commutative algebra.
Tags: discrete valuation rings, Hensel's lemma, characteristic zero
So, I’ll discuss the proof of a classification theorem that DVRs are often power series rings, using Hensel’s lemma.
Systems of representatives
The main result that we have today is:
Theorem 1 Suppose is of characteristic zero. Then , the power series ring in one variable, with respect to the usual discrete valuation on .
The “usual discrete valuation” on the power series ring is the order at zero. Incidentally, this applies to the (non-complete) subring of consisting of power series that converge in some neighborhood of zero, which is the ring of germs of holomorphic functions at zero; the valuation again measures the zero at .
For a generalization of this theorem, see Serre’s Local Fields.
To prove it, we need to introduce another concept. A system of representatives is a set such that the reduction map is bijective. A uniformizer is a generator of the maximal ideal . Then:
Proposition 2 If is a system of representatives and a uniformizer, we can write each uniquely as
Given , we can find by the definitions with . Repeating, we can write as , or . Repeat the process inductively and note that the differences tend to zero.
In the -adic numbers, we can take as a system of representatives, so we find each -adic integer has a unique -adic expansion for .
Hensel’s lemma, as already mentioned, allow us to lift approximate solutions of equations to exact solutions. This will enable us to construct a system of representatives which is actually a field.
Theorem 3 Let be a complete DVR with quotient field . Suppose and satisfies (i.e. ) while . Then there is a unique with and .
(Here the bar denotes reduction.)
The idea is to use Newton’s method of successive approximation. Recall that given an approximate root , Newton’s method “refines” it to
So define inductively ( is already defined) as , the notation as above. I claim that the approach a limit which is as claimed.
For by Taylor’s formula we can write , where depends on . Then for any
Thus, if and , we have and , since . We even have . This enables us to claim inductively:
Now it follows that we may set and we will have . The last assertion follows because is a simple root of .
There is a more general (Sorry, Bourbaki!) version of Hensel’s lemma that says if you have , the conclusion holds. It is proved using a very similar argument. Also, there’s no need for discreteness of the absolute value—just completeness is necessary.
Corollary 4 For fixed, any element of sufficiently close to 1 is a -th power.
Use the polynomial .
Proof of the Classification Theorem
We now prove the first theorem.
Note that gets sent to nonzero elements in the residue field , which is of characteristic zero. This means that consists of units, so .
Let be a subfield. Then ; if , I claim that there is containing with .
If is transcendental, lift it to ; then is transcendental over and is invertible in , so we can take .
If the minimal polynomial of over is , we have . Moreover, because these fields are of characteristic zero and all extensions are separable. So lift to ; by Hensel lift to with . Then is irreducible in (otherwise we could reduce a factoring to get one of ), so , which is a field .
So if is the maximal subfield (use Zorn), this is our system of representatives by the above argument.