TANT 9 - Non-Archimedean complete fields

Hello there! These are notes for the ninth 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.

In the previous lectures we have analyzed the set $\Sigma_K$ of places of a field $K$, and we have completely characterized it when $K$ is a number field. In order to do so we defined in the fourth lecture the notion of completion of a valued field $(K,\phi)$ and we have seen that every complete, Archimedean field is isomorphic to $(\mathbb{R},\lvert \cdot \rvert)$ or to $(\mathbb{C},\lVert \cdot \rVert)$.

So, what about the non-Archimedean case? Do we have a similar classification result? The answer is a resounding no! More precisely, we have an infinite number of non-Archimedean complete fields which are not isomorphic, as we will see by the end of the lecture.

Recall that for every non-Archimedean absolute value $\phi \colon K \to \mathbb{R}_{\geq 0}$ defined on a field $K$ we have that the unit ball $A_{\phi}$ is a local ring, i.e. a ring with a unique maximal ideal. We have also seen that if $\phi(K^{\times}) \subseteq \mathbb{R}_{> 0}$ is a discrete subgroup then $A_{\phi}$ is a discrete valuation ring, which implies in particular that the maximal ideal $\mathfrak{m}_{\phi} \subseteq A_{\phi}$ is principal.

We would like thus if for every non-Archimedean absolute value $\phi \colon K \to \mathbb{R}_{\geq 0}$ the value group $\phi(K^{\times})$ was discrete. Unfortunately this is not the case, as the following example shows.

 Example 1  Let $H \leq \mathbb{R}_{> 0}$ be any subgroup, and let $F$ be any field. We define the group ring $F[H]$ to be the set of all formal sums $x = \sum_{h \in H} x_h [h]$ where $x_h \in F$ and $x_h = 0$ for all but a finite number of $h \in H$. For $x, y \in F[H]$ we define $$x + y := \sum_{h \in H} (x_h + y_h) [h] \qquad \text{and} \qquad x \cdot y := \sum_{h \in H} \left( \sum_{h_1 h_2 = h} x_{h_1} y_{h_2} \right) [h]$$ and we see immediately that these operations turn $F[H]$ into a ring. Moreover, this ring is an integral domain. Indeed let $x, y \in F[H] \setminus \{ 0 \}$ and let $h_1 := \max\{ h \in H \mid x_h \neq 0 \}$ and analogously $h_2 := \max\{ h \in H \mid y_h \neq 0 \}$. Then clearly $(x \cdot y)_{h_1 h_2} = x_{h_1} y_{h_2} \neq 0$, because if $a b = h_1 h_2$ then either $a = h_1$ and $b = h_2$ or either $a > h_1$ or $b > h_2$. This implies that $x \cdot y \neq 0$, and thus we can define $F(H) := \operatorname{Frac}(F[H])$.

Inspired by the previous proof we can define an absolute value $\phi \colon F(H) \to \mathbb{R}_{\geq 0}$ by setting $\phi(0) = 0$ and $\phi(x) = \max\{ h \in H \mid x_h \neq 0 \}$ for all $x \in F[H]$. It is clear that $\phi(x y) = \phi(x) \phi(y)$ and that $\phi(x + y) \leq \max(\phi(x),\phi(y))$. Thus $\phi$ is a non-Archimedean absolute value and $\phi(F(H)^{\times}) = H$.

The previous exercise shows that every subgroup $H \leq \mathbb{R}_{> 0}$ can appear as a value group of some non-Archimedean absolute value $\phi \colon K \to \mathbb{R}_{\geq 0}$. Moreover, we can assume that $K$ is complete with respect to $\phi$ using the following exercise.

 Exercise 2  Let $(K,\phi)$ be a non-Archimedean valued field. Prove that its value group and its residue field don't change after completion, i.e. if $(K_{\phi},\Phi)$ is a completion of $(K,\phi)$ prove that $\phi(K^{\times}) = \Phi(K_{\phi}^{\times})$ and that $\kappa_{\phi} = \kappa_{\Phi}$. (Hint: prove that for every $x \in K_{\phi}$ we can find $a, b \in K$ satisfying $\phi(a - x) < \phi(x)$ and $\phi(b - x) < 1$).

We see thus that there is no hope to classify non-Archimedean complete fields!
Nevertheless, there is some hope in the case when the value group is discrete. Thus we give a special name to this case.

 Definition 3  Let $(K,\phi)$ be a valued field. We say that it is discretely valued if $\phi(K^{\times}) \subseteq \mathbb{R}_{> 0}$ is a discrete subgroup.

Complete, discretely valued, non-Archimedean fields

Let $(K,\phi)$ be a complete, non-Archimedean, discretely valued field. Then we know that $\mathfrak{m}_{\phi} \subseteq A_{\phi}$ is a principal ideal. This allows us to give the following definition.

 Definition 4  Let $(K,\phi)$ be a complete, non-Archimedean, discretely valued field. A uniformizer of $(K,\phi)$ is any generator of the principal ideal $\mathfrak{m}_{\phi}$.

We have already seen that if $\pi \in \mathfrak{m}_{\phi}$ is a uniformizer of a complete, non-Archimedean, discretely valued field $(K,\phi)$ we can write $x = u \pi^n$ for all $x \in K^{\times}$, where $u \in A_{\phi}^{\times}$ and $n \in \mathbb{Z}$. This allows us to show that every element of $K$ admits an expansion as a "power series" in $\pi$. For this we will need the content of the following exercise.

 Exercise 5  Let $(K,\phi)$ be a non-Archimedean field, and let $\{ x_j \}_{j \in \mathbb{N}} \subseteq K$ be any sequence. Prove that the sequence of partial sums $\left\{ \sum_{j = 0}^n x_j \right\}_n \subseteq K$ is a Cauchy sequence if and only if $\{ x_j \} \to 0$.

 Exercise 6  Let $(K, \phi)$ be any valued field, and let $\{ x_n \} \subseteq K$ be a sequence converging to $x_{\infty}$. Prove that $\phi(x_{\infty} = \lim_{n \to +\infty} \phi(x_n)$. Hint: prove first that $\phi(a - b) \geq \lvert \phi(a) - \phi(b) \rvert$ for all $a, b \in K$.

 Exercise 7  Let $(K,\phi)$ be a non-Archimedean valued field, and let $x, y \in K$ with $\phi(x) \neq \phi(y)$. Prove that $\phi(x + y) = \max(\phi(x),\phi(y))$.

 Proposition 8  Let $(K,\phi)$ be a non-Archimedean, complete, discretely valued field. For every $k \in \mathbb{N}_{\geq 1}$ let $\pi_k \in K$ be a generator of $\mathfrak{m}_{\phi}^k$ and let $S \subseteq K$ be a set of representatives for the quotient $A_{\phi} / \mathfrak{m}_{\phi}$ which contains $0$. Then we have that $$A_{\phi} = \left\{ \sum_{k = 0}^{+\infty} a_k \pi_k \mid a_k \in S \right\}$$ and if $\sum_{k = 0}^{+\infty} a_k \pi_k = \sum_{k = 0}^{+\infty} b_k \pi_k$ then $a_k = b_k$ for all $k \in \mathbb{N}$.

 Proof  Since $\mathfrak{m}_{\phi}^k = \pi^k A_{\phi} = \pi_k A_{\phi}$ we have that $\phi(\pi_k) = \phi(\pi)^k \to 0$ as $k \to +\infty$. Thus using Exercise 4 we see that for every sequence $\{ a_k \} \subseteq S$ the series $\sum_{k = 0}^{+\infty} a_k \pi_k$ converges in $K$. Using Exercise 5 and Exercise 6 we see now that $$\phi\left( \sum_{k = 0}^{+\infty} a_k \pi_k \right) = \lim_{n \to \infty} \phi\left( \sum_{k = 0}^{n} a_k \pi_k \right) = \phi(\pi_{k_0}) = \phi(\pi)^{k_0} \qquad \text{where} \qquad k_0 := \min\{ k \in \mathbb{N} \mid a_k \neq 0 \}.$$
This shows that $\sum_{k = 0}^{+\infty} a_k \pi_k \in A_{\phi}$.

Let now $x \in A_{\phi}$. We can construct inductively a sequence $\{ a_k \}_k \subseteq S$ as follows. First of all let $a_0 \in S$ be such that $x \equiv a_0 \, \text{mod} \, \mathfrak{m}_{\phi}$. Then since $\mathfrak{m}_{\phi} = \pi_1 A_{\phi}$ we can write $x = a_0 + \pi_1 x_1$ for some $x_1 \in A_{\phi}$. Now we define $a_1 \in S$ such that  $x_1 \equiv a_1 \, \text{mod} \, \mathfrak{m}_{\phi}$ and we write $x = a_0 + \pi_1 a_1 + \pi_2 x_2$. We can go on inductively by defining $a_k \in S$ to be such that $x_k \equiv a_k \, \text{mod} \, \mathfrak{m}_{\phi}$ and $x_{n + 1} := (a_0 - \pi_1 a_1 - \dots - a_{n} \pi_n)/\pi_{n + 1}$. It is thus true that $x \equiv \sum_{j = 0}^n a_j \pi_j \, \text{mod} \, \mathfrak{m}_{\phi}^{n + 1}$, which implies that the sequence of partial sums $\left\{ \sum_{j = 0}^n a_j \pi_j \right\}$ converges to $x$ as $n \to +\infty$.

Let now $\{ a_k \}, \{ b_k \} \subseteq K$ be two distinct sequences. Then we can use what we have proved in the first paragraph of this proof to show that $$\phi\left( \sum_{k = 0}^{+\infty} a_k \pi_k - \sum_{k = 0}^{+\infty} b_k \pi_k \right) = \phi(\pi_{k_0}) \neq 0 \qquad \text{where} \qquad k_0 := \min\{ k \in \mathbb{N} \mid a_k \neq b_k \}$$ which implies that $\sum_{k = 0}^{+\infty} a_k \pi_k \neq \sum_{k = 0}^{+\infty} b_k \pi_k$. Q.E.D.

 Corollary 9  Let $(K,\phi)$ be a non-Archimedean, complete, discretely valued field. Then for every $x \in K$ there exists a unique sequence $\{ a_k \}_{k = k_0}^{+\infty} \subseteq S$ (where $k_0 \in \mathbb{Z}$ and $a_{k_0} \neq 0$ ) such that $x = \sum_{k = k_0}^{+ \infty} a_k \pi^k$. Moreover $x \in A_{\phi}$ if and only if $k_0 \geq 0$.

 Proof  We know already that $x = u \pi^{k_0}$ for unique $u \in A_{\phi}^{\times}$ and $k_0 \in \mathbb{Z}$. Moreover we can use the previous proposition with $\pi_k = \pi^k$ to show that there exists a unique sequence $\{ b_j \}_{j = 0}^{+\infty} \subseteq S$ such that $u = \sum_{j = 0}^{+\infty} b_j \pi^j$. The fact that $\phi(u) = 1$ gives us that $b_0 \neq 0$. Thus we have that $x = \sum_{k = k_0}^{+ \infty} a_{j} \pi^k$ where $a_j := b_{j - k_0}$. Q.E.D.

Local fields

Until now, we have only dealt with valued fields $(K,\phi)$ but, as we have seen, these are special cases of topological fields.

 Definition 10  A topological field is a field $K$ which is also a topological space such that the maps $$\begin{aligned}     K \times K &\to K \\     (x,y) &\mapsto x + y     \end{aligned} \qquad \text{and} \qquad \begin{aligned}     K \times K &\to K \\     (x,y) &\mapsto x y     \end{aligned} \qquad \text{and} \qquad     \begin{aligned}     K^{\times} &\to K^{\times} \\ x &\mapsto x^{-1}     \end{aligned}$$ are continuous.

Among all topological fields we need to pay special attention to the ones which are locally compact.

 Definition 11  A topological space $X$ is said to be locally compact if for every point $x \in X$ there exist an open subset $U \subseteq X$ and a compact subset $C \subseteq X$ such that $x \in U \subseteq C$.

We will see later on that topological fields which are locally compact are automatically valued fields, i.e. the topology comes from an absolute value $\phi \colon K \to \mathbb{R}_{\geq 0}$. This absolute value comes from the Haar measure that we can define on $K$ thanks to the fact that it is locally compact. For this reason, locally compact fields deserve a special name.

 Definition 12  A local field $K$ is a topological field which is locally compact as a topological space.

Even if we cannot prove at the moment that every local field is a valued field, we can prove a nice classification result about complete, non-Archimedean valued fields which are locally compact. In order to do so we need to introduce the field of formal Laurent series over a given field.

 Example 13  Let $F$ be a field. We can define the ring of formal power series with coefficients in $F$ as $$F[[ T ]] := \left\{ \sum_{j = 0}^{+\infty} a_j T^j \mid a_j \in F \right\}$$ with the usual operations of sum and product of series, namely $$\sum_{j = 0}^{+\infty} a_j T^j + \sum_{j = 0}^{+\infty} b_j T^j := \sum_{j = 0}^{+\infty} (a_j + b_j) T^j \qquad \text{and} \qquad \sum_{j = 0}^{+\infty} a_j T^j \cdot \sum_{j = 0}^{+\infty} b_j T^j := \sum_{j = 0}^{+\infty} \left( \sum_{k + l = j} a_k b_l \right) T^j.$$
One can prove quite easily that $F[[ T ]]$ is an integral domain and that its field of fraction is the field of formal Laurent series $$F(( T )) := \left\{ \sum_{j = j_0}^{+\infty} a_j T^j \mid a_j \in F, \, j_0 \in \mathbb{Z} \right\}$$ with the same operations defined above.

We will see in the next lecture how this field arises as a completion of the field $F(T)$ of rational functions with coefficients in $F$ with respect to the absolute value $\lvert \cdot \rvert_{t}$ that we defined in the second lecture. We will use this result in the proof of the following theorem.

 Theorem 14  Let $(K,\phi)$ be a non-trivial valued field and suppose that $\mathcal{T}_{\phi}$ is locally compact. Then $(K,\phi)$ is complete. Moreover, if $\phi$ is Archimedean then $K \cong \mathbb{R}$ or $K \cong \mathbb{C}$ and if $\phi$ is non-Archimedean then $(K,\phi)$ is either a finite extension of $\mathbb{Q}_p$ or a finite extension of $\mathbb{F}_p((T))$

 Proof  Observe first of all that every closed ball $B_{\varepsilon}(x) := \{ y \in K \mid \phi(y -x) \leq \varepsilon \}$ is compact. Indeed since $K$ is locally compact there exists $\delta \in \mathbb{R}_{> 0}$ such that $B_{\delta}(0)$ is compact. Let now $z \in K$ be any element with $\phi(z) > 1$, which exists because $\phi$ is not trivial. Then the balls $B_{\phi(z)^n \delta}(0)$ are compact for all $n \in \mathbb{N}$ since $B_{\phi(z)^n \delta}(0) = (\mu_z)^n(B_{\delta}(0))$, where $\mu_z \colon K \to K$ is the (continuous) multiplication map defined by $\mu_z(y) := z \cdot y$. Thus for every $\varepsilon > 0$ the ball $B_{\varepsilon}(0)$ is compact because it is a closed subspace of $B_{\phi(z)^n \delta}(0)$ for $n$ sufficiently big. Finally we only need to observe that $B_{\varepsilon}(x) = \mathfrak{t}_x(B_{\varepsilon}(0))$ for all $x \in K$, where $\mathfrak{t}_x \colon K \to K$ is the (continuous) translation map defined by $\mathfrak{t}_x(y) = x + y$.

The compactness of the closed balls implies immediately that $K$ is complete. Indeed let $\{ x_n \} \subseteq K$ be any Cauchy sequence. Since this sequence is Cauchy we can find $n_0 \in \mathbb{N}$ such that $x_n \in B_1(x_{n_0})$ for all $n \geq n_0$. Thus the Cauchy sequence $\{ y_n \} \subseteq K$ defined by $y_n := x_{n + n_0}$ is convergent in $B_1(x_{n_0})$ (because every compact metric space is complete, see here), which clearly implies that $\{ x_n \}$ converges in $K$.

Now, if $\phi$ is Archimedean then we can use Theorem 16 of the fourth lecture to conclude that $K$ is isomorphic either to $\mathbb{R}$ or to $\mathbb{C}$. Suppose now that $\phi$ is non-Archimedean. Then if $\operatorname{char}(K) = 0$ we have an embedding $\mathbb{Q} \hookrightarrow K$ and the absolute value $\phi$ restricts to a non-Archimedean absolute value on $\mathbb{Q}$. Now we can use Theorem 7 of the third lecture to see that this restriction must be equivalent to $\lvert \cdot \rvert_p$ for some prime $p \in \mathbb{N}$. Using the fact that $\mathbb{Q}_p$ is the completion of $\mathbb{Q}$ with respect to $\lvert \cdot \rvert_p$ (which we will prove in the next lecture) we obtain an embedding $\mathbb{Q}_p \hookrightarrow K$. To prove that the degree of this extension is finite we will need to use the fact that if $F$ is a locally compact topological field, and $V$ is a topological vector space over $F$ which is locally compact then $V$ is finite dimensional (Terence Tao proves this when $F = \mathbb{R}$, but the proof generalizes to the general case).

Suppose now that $\operatorname{char}(K) = p > 0$. This implies that we get an embedding $\mathbb{F}_p \hookrightarrow K$. Let now $\pi \in K$ be a uniformizer. Then $\pi$ is transcendental over $\mathbb{F}_p$, because if this was algebraic then it would be finite, and thus $\mathbb{F}_p(\pi)$ would be a finite field, but Exercise 8 of the second lecture tells us that every valuation on a finite field is trivial, whereas we have that $\phi(\pi) < 1$. Thus we have that $\mathbb{F}_p(\pi)$ is isomorphic to the field of rational functions $\mathbb{F}_p(T)$ and we get an embedding $\mathbb{F}_p(T) \hookrightarrow K$. This implies (as before) that $\phi$ restricts to an absolute value $\psi$ on $\mathbb{F}_p(T)$ with $\psi(T) < 1$. But we have only one such absolute value, namely $\lvert \cdot \rvert_T$, and thus we get an embedding $\mathbb{F}_p((T)) \hookrightarrow K$ because (as we will prove in the next lecture) $\mathbb{F}_p((T))$ is (isomorphic to) the completion of $\mathbb{F}_p(T)$ with respect to $\lvert \cdot \rvert_T$. We can apply again the theorem on the finite dimensionality of locally compact vector spaces to prove that this is a finite extension. Q.E.D.

A simple consequence of the previous theorem is that every non-Archimedean valued field $(K,\phi)$ such that $\mathcal{T}_{\phi}$ is locally compact is also complete and its residue field $\kappa_{\phi}$ is finite. It is quite easy to prove that the following also holds.

 Proposition 15  Let $(K,\phi)$ be a non-Archimedean, complete field whose residue field $\kappa_{\phi}$ is finite. Then $\mathcal{T}_{\phi}$ is locally compact.

 Proof  We prove first of all that $A_{\phi}$ is compact. Indeed let $\{ U_{\alpha} \}_{\alpha \in \mathcal{A}}$ be an open cover of $A_{\phi}$ and suppose by contradiction that it does not admit a finite subcover. Observe now that $$A_{\phi} = (x_1 + \pi A_{\phi}) \cup \dots \cup (x_n + \pi A_{\phi})$$ where $\pi \in \mathfrak{m}_{\phi}$ is a uniformizer and $\{ x_1,\dots,x_n \} \subseteq A_{\phi}$ are representatives for the finite quotient $A_{\phi}/\mathfrak{m}_{\phi}$. Since $\{ x_1,\dots,x_n \}$ are finite there exists $j_0 \in \{ 1,\dots,n \}$ such that $x_{j_0} + \pi A_{\phi}$ is not covered by a finite number of $U_{\alpha}$'s. Since we have that $$x_{j_0} + \pi A_{\phi} = x_{j_0} + \pi x_1 + \pi^2 A_{\phi} \cup \dots \cup x_{j_0} + \pi x_n + \pi^2 A_{\phi}$$ there exists $j_1 \in \{ 1,\dots,n \}$ such that $x_{j_0} + \pi x_{j_1} + \pi^2 A_{\phi}$ is not covered by a finite number of $U_{\alpha}$'s. Going on like this we can always find $j_k \in \{ 1,\dots,n \}$ such that $x_{j_0} + \dots + \pi^k x_{j_k} + \pi^{k+1} A_{\phi}$ is not covered by a finite number of $U_{\alpha}$'s. Moreover the series $\sum_{k = 0}^{+\infty} x_{j_k} \pi^k$ converges (why?) and we have that $x := \sum_{k = 0}^{+\infty} x_{j_k} \pi^k \in A_{\phi}$. Thus $x \in U_{\alpha}$ for some $\alpha \in \mathcal{A}$ and since $U_{\alpha}$ is open this implies that $x + \pi^n A_{\phi} \subseteq U_{\alpha}$ for some $n \in \mathbb{N}$, because $\{ x + \pi^n A_{\phi} \}_n$ is a system of open neighborhoods of $x$. But this is a contradiction, because it would imply that $x_{j_0} + \dots + \pi^{n - 1} x_{j_{n - 1}} + \pi^n A_{\phi}$ is contained in a finite number of $U_{\alpha}$'s.

To conclude we only need to observe that for every point $x \in K$ we have that $x \in x + \pi A_{\phi} \subseteq x + A_{\phi}$ where $x + \pi A_{\phi}$ is clearly open and $x + A_{\phi}$ is clearly compact. Q.E.D.

Conclusions and references

In this lecture we managed to:

• prove that every element of a non-Archimedean, complete, discretely valued field $(K,\phi)$ can be written in a unique way as a power series in the uniformizer $\pi \in K$;
• give a characterization of local fields as finite extensions of $\mathbb{R}$, $\mathbb{Q}_p$ and $\mathbb{F}_p((T))$.

References for this lecture include:

• Section 2 of these notes by Stevenhagen;
• these notes by Clark;
• these notes by Sutherland;
• these notes by Bouyer.