In the last post, we studied the algebraic definition of p-adic numbers and Hensel’s lemma. In Part III here, we are study Krasner’s lemma and its applications. Our goal is to understand how analysis governs algebra in the -adic world, which is quite different from the archimedean world.

Before we start, we’ll briefly study about -adic norms on finite extensions of . Let’s start with a simplest nontrivial example: a quadratic extension. Recall that the equation has no zero in . So we can consider degree 2 extension of , , by adjoining . Now, we want to extend our 5-adic norm to . The only condition we’ll impose is the following: we want the extended norm is invariant under conjugation. More precisely, in the case of , we want to have for all . This implies

and one can show that this actually defines a non-archimedean norm on . More generally, for a given irreducible polynomial and its root , we define its -adic norm as

(Here is determinant of the multiplication map , which is just .) So -adic norms of elements has a form with .

Our main theorem for this post is the following:

**Theorem (informal).** For two given irreducible polynomials , splitting fields of over coincide when and are sufficiently close.

To prove this, we first need to define what “sufficiently close polynomials” means.

**Definition 1.** For a given polynomial , we define its norm as

**Theorem 1.** Let be a polynomial in . If is a zero of , we have

*Proof.* Indeed, there’s nothing to prove for , and if , we have

which proves the inequality.

Now we will show that the zeros of a polynomial over are continuous functions on its coefficients.

**Theorem 2.** Let be a polynomial of degree with distinct zeros in . If a polynomial of degree has all coefficients sufficiently close to those of , i.e. if is sufficiently small, then it has roots which approximate the roots to sufficiently high precision.

Before proving the theorem, we note that the theorem is false for polynomials over . In fact, if we consider , its root is 0, but zeros of aren’t even in for .

*Proof*. Let and . Choose so that

where

Suppose that satisfies , and let be any root of . Then gives . So we get . Thus

Therefore , hence one of the factors must be smaller than . This shows that is within of a root of . Conversely, we can prove that for any zero of , there exists a zero of very close to by the same argument (using and ).

To prove our main theorem, we need one more crucial “lemma” : Krasner’s lemma.

**Lemma 1 (Krasner)**. Let and let be its conjugates over . If and

(which means that is closer to than any other its conjugates) then .

*Proof*. Assume that . Then is a field extension of degree . So we have a field embedding such that , but . Then for some , and since the -adic norm is invariant under the conjugation

However, this implies

which contradicts our assumption.

From the above theorems and Krasner’s lemma, we can finally prove our main theorem which illustrates how analysis can govern algebra in -adic fields.

**Theorem 3** Let be an irreducible polynomial and let be a zero of . Then there exists such that for all with , there exists a zero of such that . In particular, is also irreducible.

*Proof:* By Theorem 2, there exists such that any polynomial with satisfies

for all , where are zeros of . Then Krasner’s lemma gives .

Using these theorems, we can prove the following:

**Theorem 4.** For any , the algebraic closure of is not complete under the -adic metric extended to . However, its completion is algebraically closed.

*Proof:* For the first statement, consider the sequence

It is clear that , and also the sequence is a Cauchy sequence since converges to 0. However, the limit does not exist in . Indeed, let be any element in which has degree over . Then norms of each zeros should be a form of for some . In other words, denominators of the exponents in the norm are bounded by . However, it is not hard to check that for , which tends to infinity as grows. Thus does not exist in .

To prove the second statement, let be a zero of some monic polynomial . We want to show that . By the Theorem 3, we can find such that is sufficiently small so for some zero of . This proves .

In fact, the above theorem is true for any field which is complete with respect to some non-trivial non-Archimedean absolute value . See [2] for the proof.

In the next post, we will study further about finite (and infinite?) extensions of , such as their Galois groups.

**References. **

[1] Brian Conrad, *Completion of Algebraic Closure*, http://virtualmath1.stanford.edu/~conrad/248APage/handouts/algclosurecomp.pdf, Online note.

[2] Brian Conrad, *Higher ramification groups*, http://math.stanford.edu/~conrad/676Page/handouts/ramgroup.pdf, Online note.

[3] Jürgen Neukirch, Algebraic number theory. Vol. 322. Springer Science & Business Media, 2013.