Topologies determine the absolute value

Time to go back to basic algebraic number theory (which we’ll need for two of my future aims here: class field theory and modular representation theory), and to throw in a few more facts about absolute values and completions—as we’ll see, extensions in the complete case are always unique, so this simplifies dealing with things like ramification. Since ramification isn’t affected by completion, we can often reduce to the complete case.

Absolute Values

Henceforth, all absolute values are nontrivial—we don’t really care about the absolute value that takes the value one everywhere except at zero.

I mentioned a while back that absolute values on fields determine a topology. As it turns out, there is essentially a converse.

Theorem 1 Let ${\left|\cdot\right|_1}$ , ${\left|\cdot\right|_2}$ be absolute values on ${K}$ inducing the same topology. Then ${\left|\cdot\right|_2}$ is a power of ${\left|\cdot\right|_1}$ .

So, first of all, given ${x \in K^*}$ , the sequence ${\{x^n\}}$ converges to zero iff ${\left|x\right|_1<1}$ , and iff ${\left|x\right|_2 < 1}$ . Thus we see that the unit disks are the same under both absolute values.

Now fix ${x}$ with ${\left|x\right|_1<1}$ and consequently ${\left|x\right|_2<1}$ , by the previous paragraph. Suppose ${\alpha>0}$ and for any ${y}$ ,

$\displaystyle \left|x\right|_1 \leq \left|y\right|_1^{\alpha};\ \ \ \ \ (1)$

we have to show

$\displaystyle \left|x\right|_2 \leq \left|y\right|_2^{\alpha}\ \ \ \ \ (2)$

as we can then use symmetry to show that ${\frac{ \log \left|y\right|_1}{\log \left|x\right|_1} = \frac{ \log \left|y\right|_2}{\log \left|x\right|_2}}$ , and then one gets that ${\left|y\right|_1}$ , ${\left|y\right|_2}$ differ by the same power as the ${x}$ ‘s.

First, I claim this is true for ${\alpha \in \mathbb{N}}$ . Then (1) says

$\displaystyle \left|x/y^{\alpha}\right|_1 < 1$

so (2) follows in this case by the first paragraph. By raising each side to an integer power, it follows that (1) implies (2) for ${\alpha \in \mathbb{Q}^+}$ , and by continuity for ${\alpha >0}$ arbitrary.

Complete Fields

Next, I claim that in the complete case, absolute values are very often going to be equivalent.

So fix a field ${K}$ complete with respect to the absolute value ${\left|\cdot\right|}$ . Let ${V}$ be a finite-dimensional vector space over ${K}$ . A norm on ${V}$ isa map ${\left\lVert{ \cdot}\right\rVert: V \rightarrow \mathbb{R}_{\geq 0}}$ satisfying all the usual constraints, i.e. scalar multiplicativity (with respect to ${\left|\cdot\right|}$ ) and the triangle inequality.

Theorem 2 All norms on ${V}$ induce the same topology.

For the proof, I’ll make the simplifying (but unnecessary) assumption that ${K}$ is locally compact—this is the case for the real and complex numbers, finite extensions of the ${p}$ -adic fields, and finite extensions of power series fields ${\mathbb{F}_q((X))}$ (all of which go under the name “local fields”).

Fix a basis ${v_1, \dots, v_n \in V}$ . We can choose the norm ${\left\lVert{\cdot}\right\rVert_1}$ on ${V}$ : if ${v = \sum c_i v_i}$ , then ${\left\lVert{v}\right\rVert_1 = \max \left|c_i\right|}$ . Then from the triangle inequality it is easy to check that for all ${v}$ , ${\left\lVert{v}\right\rVert \leq C \left\lVert{v}\right\rVert_1}$ for some large ${C}$ . If we show the reverse, that there exists a ${c>0}$ with ${\left\lVert{v}\right\rVert \geq c \left\lVert{v}\right\rVert_1}$ for all ${v}$ , then the assertion will be complete. By homogeneity we may assume ${\left\lVert{v}\right\rVert_1 = 1}$ ; the set of such ${v}$ forms a compact subset of ${V}$ (under the ${\left\lVert{v}\right\rVert_1}$ -topology!), on which by the previous inequality ${\left\lVert{\cdot}\right\rVert}$ is a continuous and nonzero function. It has a minimum, which is the ${c}$ referred to.

We can apply this to the case of fields:

Corollary 3 Given a complete field ${K}$ and a finite extension ${L \supset K}$ , the absolute value on ${K}$ can be extended to ${L}$ in precisely one way.

The above result implies uniqueness; we’ve shown existence earlier, at least in the discrete case. So in particular if ${A}$ is a complete DVR and ${B}$ the integral closure in some extension of the quotient field of ${A}$ of degree ${n}$ , and if ${\mathfrak{P} \subset B}$ prolongs ${\mathfrak{p} \subset A}$ , then the basic ramification formula becomes