TANT 2 - Absolute values

Hello there! These are notes for the second class of the course "Topics in algebra and number theory" held in Block 4 of the academic year 2017/18 at the University of Copenhagen.

As we said last time, the p-adic numbers $\mathbb{Q}_p$ are obtained with a process of completion analogous to the classical way in which we obtain the real numbers $\mathbb{R}$ by "filling the holes" in the rational numbers $\mathbb{Q}$. In order to obtain $\mathbb{Q}_p$ we need to introduce a new notion of distance (and, thus, new "holes") on the rational numbers, which will be related to the prime $p$. This can be done in every field as we see in the next section.

Absolute values on fields

We give first of all some definitions of the correct generalization of the standard absolute value $\mathbb{R} \to \mathbb{R}_{\geq 0}$ to any field.

 Definition 1  Let $K$ be any field. A weak absolute value on $K$ is a map $\phi \, \colon K \to \mathbb{R}_{\geq 0}$ such that:

• $\phi(x) = 0$ if and only if $x = 0$;
• $\phi(x y) = \phi(x) \phi(y)$ for all $x,y \in K$;
• there exists a constant $C \in \mathbb{R}_{>0}$ such that $\phi(x + y) \leq C \max(\phi(x),\phi(y))$ for all $x,y \in K$.

 Definition 2  We define the norm of a weak absolute value $\phi \colon K \to \mathbb{R}_{\geq 0}$ to be $$\lVert \phi \rVert := \inf\{ C \in \mathbb{R}_{>0} \mid \phi(x + y) \leq C \max(\phi(x),\phi(y)) \ \text{for all} \ x,y \in K  \} \in \mathbb{R}_{\geq 0}$$ which exists thanks to the third part of the previous definition.

 Definition 3  We say that $\phi$ is an absolute value if $\lVert \phi \rVert \leq 2$.

 Definition 4  We say that an absolute value $\phi$ is non-Archimedean if $\lVert \phi \rVert = 1$.We say that it is Archimedean otherwise.

These definitions need to be explained a little bit. First of all, observe that a weak absolute value $\phi \colon K \to \mathbb{R}_{\geq 0}$ is an extension of a multiplicative map $K^{\times} \to \mathbb{R}_{> 0}$ to the whole $K$ by setting $\phi(0) = 0$.
Moreover the third part of Definition 1 says that $\phi$ is "continuous with respect to the sum", i.e. the absolute value of the sum of two elements of $K$ must not be far from the absolute values of the two elements themselves.

So, why is this only a weak absolute value? The reason is that we want absolute values to be a measure of distances between two elements of a field. In particular, as we will see in a minute, we want every absolute value $\phi$ to define a metric on the set of elements of $K$ by setting $$\begin{aligned} K \times K &\to \mathbb{R}_{\geq 0}. \\ (x, y) &\mapsto \phi(x - y) \end{aligned} .$$
For this to be true we need $\phi$ to satisfy the triangle inequality $\phi(x + y) \leq \phi(x) + \phi(y)$ and the following lemma explains that this happens precisely when $\lVert \phi \rVert \leq 2$.

 Lemma 5  Let $\phi \colon K \to \mathbb{R}_{\geq 0}$ be a weak absolute value. Then $\lVert \phi \rVert \leq 2$ if and only if $\phi(x+y) \leq \phi(x) + \phi(y)$ for all $x,y \in K$.

 Proof  Clearly if $\phi(x+y) \leq \phi(x) + \phi(y)$ then in particular $\phi(x+y) \leq 2 \max(\phi(x),\phi(y))$ and so $\lVert \phi \rVert \leq 2$. The converse is a little more complicated.

Suppose that $\phi(x + y) \leq 2 \max(\phi(x),\phi(y))$ for all $x,y \in K$. Then by applying this repeatedly we get that $$\phi\left( \sum_{j = 1}^{2^r} x_j \right) \leq 2 \max\left(\phi\left( \sum_{j = 1}^{2^{r-1}} x_j \right), \phi\left( \sum_{j = 2^{r - 1}+1}^{2^r} x_j \right) \right) \leq \dots \leq 2^r \max_{j}(\phi(x_j))$$ for all $x_1,\dots,x_{2^r} \in K$. Using this we get that for every $x_1,\dots, x_n \in K$ we have $$\phi\left( \sum_{j = 1}^n x_j \right) \leq 2 \, n \max_{j}(\phi(x_j))$$ because if $2^{r - 1} \leq n \leq 2^r$ then $2 n \geq 2^r$, and we can use the previous inequality with $x_{n+1} = \dots = x_{2^r} = 0$. In particular we get that $\phi(n \, x) \leq 2 \, n \, \phi(x)$ for all $x \in K$.

We can apply now these inequalities to $(\phi(x+y))^n$, and we get $$(\phi(x+y))^n = \phi((x+y)^n) = \phi\left( \sum_{j = 0}^n \binom{n}{j} x^j y^{n - j} \right) \leq 2 \, (n+1) \, \max_j\left\{ \phi\left( \binom{n}{j} \right) \, \phi(x)^j \, \phi(y)^{n - j} \right\} = \dagger$$ and now if we use the fact that $\phi(k) \leq 2 \, k$ for all $k \in \mathbb{Z}$ and we sum over $j$ instead of taking the maximum we get $$\dagger \leq 4 \, (n + 1) \, \max_j\left\{ \binom{n}{j} \phi(x)^j \phi(y)^{n - j} \right\} \leq 4 \, (n + 1) \, \sum_{j = 0}^{n} \binom{n}{j} \phi(x)^j \phi(y)^{n - j} = 4 \, (n + 1) \, (\phi(x) + \phi(y))^n$$ which implies that $\phi(x+y) \leq ( 4 \, (n + 1) )^{1/n} \, (\phi(x) + \phi(y))$ and thus by taking the limit for $n \to +\infty$ we get the triangle inequality. Q.E.D.

Finally we need to explain why weak absolute values $\phi$ with $\lVert \phi \rVert = 1$ are called non-Archimedean. Recall that a totally ordered abelian group $G$ is said to satisfy the Archimedean property if for every $x, y \in G$ with $x,y \geq 0$ there exists $n \in \mathbb{N}$ such that $n \, x \geq y$. Putting this in simpler terms, we want that for any positive element $x \in G$ the elements $n \, x$ become bigger and bigger as $n \to +\infty$. Since in our case $\phi$ measures "how big elements are" we see immediately that if  $\lVert \phi \rVert = 1$ the Archimedean property does not hold anymore. More precisely, $$\phi(2 \, x) = \phi(x + x) \leq \max(\phi(x),\phi(x)) = \phi(x)$$ and in general $\phi(n \, x) \leq \phi(x)$ for all $x \in K$.

 Exercise 6  Show that a weak absolute value $\phi \colon K \to \mathbb{R}_{> 0}$ is non-Archimedean if and only if the set $\{ \phi(n) \}_{n \in \mathbb{Z}} \subseteq \mathbb{R}$ is bounded. Deduce as a corollary that every (weak) absolute value on a field of positive characteristic is non-Archimedean. (Hint: use the same argument contained in the proof of Lemma 5).

Examples of absolute values

We give some examples of absolute values, particularly in the case of number fields, upon which we will return in the third lecture.

 Example 7  On any field $K$ we have the trivial absolute value $\lvert \cdot \rvert_t \colon K \to \mathbb{R}_{\geq 0}$ simply defined by $\lvert x \rvert_t = 1$ for all $x \in K^{\times}$ and $\lvert 0 \rvert_t = 0$.

 Exercise 8  Prove that every absolute value on a finite field is trivial.

 Example 9  We have the two canonical absolute values $$\begin{aligned} \lvert \cdot \rvert \colon \mathbb{R} &\to \mathbb{R}_{\geq 0} \\ x &\mapsto \begin{cases} x, \, \text{if} \, x \geq 0 \\ -x, \, \text{if} \, x \leq 0 \end{cases} \end{aligned} \qquad \text{and} \qquad \begin{aligned} \lvert \cdot \rvert \colon \mathbb{C} &\to \mathbb{R}_{\geq 0} \\ x  &\mapsto \sqrt{\Re(x)^2 + \Im(x)^2} \end{aligned}$$

 Example 10  Let $R$ be a Dedekind domain with field of fractions $K$ and let $\mathfrak{p} \subseteq R$ be a prime ideal. We can use the fact that $K = \operatorname{Frac}(R)$ to define a weak absolute value $\lvert \cdot \rvert_{\mathfrak{p}}$ on $K$ by putting $\lvert x \rvert_{\mathfrak{p}} := c^{- \operatorname{ord}_{\mathfrak{p}}(x)}$ for all $x \in R$, where $c \in \mathbb{R}_{> 1}$ is some chosen constant. Here we use the unique factorization of ideals in $R$ to define $\operatorname{ord}_{\mathfrak{p}}(x) \in \mathbb{N}$ to be the unique natural number such that $x \, R = \mathfrak{p}^{\operatorname{ord}_{\mathfrak{p}}(x)} \cdot \mathfrak{a}$ with $\mathfrak{p} \nmid \mathfrak{a}$. We can in particular think that $\operatorname{ord}_{\mathfrak{p}}(0) = +\infty$ and thus $\lvert 0 \rvert_{\mathfrak{p}} = 0$.

Observe that $\lvert \cdot \rvert_{\mathfrak{p}}$ is non-Archimedean. Indeed this is equivalent to say that $\operatorname{ord}_{\mathfrak{p}}(x+y) \geq \min(\operatorname{ord}_{\mathfrak{p}}(x),\operatorname{ord}_{\mathfrak{p}}(y))$ for all $x, y \in R$, which is true because if $x R = \mathfrak{p}^n \, \mathfrak{a}$ and $y R = \mathfrak{p}^m \, \mathfrak{b}$ for some ideals $\mathfrak{a}, \mathfrak{b} \subseteq R$ then we have $(x + y) R = \mathfrak{p}^{\min(n,m)} \mathfrak{c}$ for some ideal $\mathfrak{c} \subseteq R$.

We will in particular apply this construction when $R = \mathcal{O}_K$ is the ring of integers of some number field $K$ and when $R = F[x]$ is the ring of polynomials over a field $F$.

 Example 11  Let $K \hookrightarrow L$ be a field extension and let $\lvert \cdot \rvert_L \colon L \to \mathbb{R}_{\geq 0}$ be an absolute value. Then clearly the restriction of $L$ to $K$ gives us another absolute value $\lvert \cdot \rvert_K \colon K \to \mathbb{R}_{\geq 0}$.
We will in particular apply the previous construction to the embeddings $\sigma \colon K \hookrightarrow \mathbb{C}$ of a number field $K$ inside $\mathbb{C}$, and we will denote the corresponding absolute value by $\lvert \cdot \rvert_{\sigma}$.

 Example 12  Let $K = F(t)$ for some field $F$. Then we have the absolute value $\lvert \cdot \rvert_{\infty}$ which for a rational function $f(t) = P(t)/Q(t) \in K^{\times}$ is defined to be $\lvert f(t) \rvert_{\infty} := c^{\deg(P) - \deg(Q)}$ for some constant $c \in \mathbb{R}_{> 1}$.

The topology induced by a weak absolute value and equivalence of weak absolute values

Let $\phi \colon K \to \mathbb{R}_{\geq 0}$ be a weak absolute value. Then $\phi$ induces a topology $\mathcal{T}_{\phi}$ on $K$. A basis of open sets for this topology is given by the subsets $U_{\varepsilon}(x) := \{ y \in K \, \mid \, \phi(y - x) < \varepsilon \} \subseteq K$ for all $x \in K$ and $\varepsilon \in \mathbb{R}_{> 0}$.

 Exercise 13  Prove that if $\phi \, \colon \, K \to \mathbb{R}_{\geq 0}$ is a weak absolute value then the sum and product maps $K \times K \to K$ and the inversion map $K^{\times} \to K^{\times}$ are continuous with respect to the product topology on $K \times K$ and the subspace topology on $K^{\times}$. This is equivalent to say that $(K, \mathcal{T}_{\phi})$ is a topological field.

 Exercise 14  Prove that $\mathcal{T}_{\phi}$ is the discrete topology if and only if $\phi$ is trivial and that $\mathcal{T}_{\phi}$ is metrizable if and only if $\phi$ is an absolute value.

We are in particular interested in the topological properties of the field $K$ and not really in how the weak absolute value $\phi$ was defined in the first place. For this reason we give the following definition.

 Definition 15  We say that two weak absolute values are equivalent if and only if they induce the same topology on $K$.

We have two nice and concrete characterizations of equivalence given by the following theorem.

 Theorem 16  Let $\phi, \psi \colon K \to \mathbb{R}_{\geq 0}$ be two non trivial weak absolute values. Then the following properties are equivalent:

1. There exists $r \in \mathbb{R}_{> 0}$ such that $\phi(x) = \psi(x)^r$ for all $x \in K$;
2. $\mathcal{T}_{\phi} = \mathcal{T}_{\psi}$, i.e. $\phi$ and $\psi$ are equivalent;
3. $\phi(x) < 1$ if and only if $\psi(x) < 1$ for all $x \in K$

 Proof  It is clear that if 1. holds then $U_{\varepsilon}^{\psi}(x) = U_{\varepsilon^r}^{\phi}(x)$ for all $x \in K$ and $\varepsilon \in \mathbb{R}_{> 0}$, and thus 2. holds.

Suppose now that 2. holds and let $x \in K$ be such that $\phi(x) < 1$. This is equivalent to say that the sequence $\{ \phi(x)^n = \phi(x^n) \}_n$ goes to zero as $n \to +\infty$, which also equivalent to say that the sequence $\{ x^n \}_n$ "goes to zero" in the topology $\mathcal{T}_{\phi}$, i.e. for all $\varepsilon \in \mathbb{R}_{> 0}$ there exists $n_0 \in \mathbb{N}$ such that $x_n \in U_{\varepsilon}^{\phi}(0)$ for all $n \geq n_0$. But now we know that for all $\varepsilon \in \mathbb{R}_{> 0}$ there exist $\delta_1, \delta_2 \in \mathbb{R}_{>0}$ such that $U_{\delta_1}^{\psi}(0) \subseteq U_{\varepsilon}^{\phi}(0) \subseteq U_{\delta_2}^{\psi}(0)$. This implies in particular that $\{ x^n \}_n$ "goes to zero" also in the topology $\mathcal{T}_{\psi}$ and thus that $\psi(x) < 1$. The converse is proved in the same way and thus 3. holds.

Suppose finally that 3. holds. Observe first of all that $\phi(1) = \phi(1)^2 = 1$ and thus $\phi(x^{-1}) = \phi(x)^{-1}$. For this reason for every $x \in K$ we have that either $\phi(x) < 1$ or $\phi(x^{-1}) < 1$ or $\phi(x) = 1$. Since we are assuming that $\phi$ is not trivial we can thus find $a \in K$ such that $\phi(a) < 1$. By assumption we also have $\psi(a) < 1$. Define now two functions $\alpha(x), \beta(x) \colon K^{\times} \to \mathbb{R}$ by setting $$\alpha(x) := \frac{\log(\phi(x))}{\log(\phi(a))} \qquad \text{and} \qquad \beta(x) := \frac{\log(\psi(x))}{\log(\psi(a))}$$ for all $x \in K^{\times}$. Observe now that if $m \in \mathbb{Z}$ and $n \in \mathbb{N}_{\geq 1}$ we have that $$\frac{m}{n} < \alpha(x) \iff \phi(x) = \phi(a)^{\alpha(x)} < \phi(a)^{m/n} \iff \phi(x^n a^{-m}) \leq 1 \iff \psi(x^n a^{-m}) \leq 1 \iff \frac{m}{n} < \beta(x)$$ and thus we get $\alpha(x) = \beta(x)$ for all $x \in K$. Thus the real number $$r := \frac{\log(\phi(x))}{\log(\psi(x))} = \frac{\alpha(x)}{\beta(x)} \, \frac{\log(\phi(a))}{\log(\psi(a))} = \frac{\log(\phi(a))}{\log(\psi(a))}$$ does not depend on $x$ and we get that $\phi(x) = \psi(x)^r$ for all $x \in K$. Thus 1. holds and we have finished the proof. Q.E.D.

In view of the previous theorem we will only consider weak absolute values up to equivalence. Observe in particular that every weak absolute value is equivalent to a (proper) absolute value by scaling it in such a way that $\phi(2) \leq 2$, in view of the following exercise.

 Exercise 17  Let $\phi \colon K \to \mathbb{R}_{\geq 0}$ be a weak absolute value. Prove that $$\lVert \phi \rVert = \sup_{\substack{t \in K \\ \phi(t) \leq 1}} \phi(1 + t)$$ and moreover that $\lVert \phi \rVert = \max(\phi(2),1)$.

For this reason from the following lecture onward we will only consider (equivalence classes of) absolute values, which deserve a special name.


 Definition 18  Let $K$ be any field. The set of places of $K$ the defined as the set $\Sigma_{K}$ of absolute values $K \to \mathbb{R}_{\geq 0}$ up to equivalence.

 Exercise 19  Prove that if $\phi, \psi \colon K \to \mathbb{R}_{\geq 0}$ are two (weak) equivalent absolute values then $\phi$ is non-Archimedean if and only if $\psi$ is non-Archimedean.

In view of the previous exercise and of Exercise 14 we can write $\Sigma_K = \Sigma_K^{\infty} \sqcup \Sigma_{K,\infty} \sqcup \{ \lvert \cdot \rvert_t \}$, where $\Sigma_K^{\infty}$ is the set of equivalence classes of non-Archimedean (and non-trivial) absolute values, and $\Sigma_{K,\infty}$ is the set of equivalence classes of the Archimedean ones.

Conclusions and references

In this class we have defined weak absolute values on a field $K$, we have seen when they respect the triangle inequality and we have defined an equivalence relation between them by looking at the topologies that they induce on $K$.

The main reference for this class is Section 1 of Stevenhagen's notes (from which we have borrowed quite a lot if not all the material in this post!).