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.
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.
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.
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 \( \mathbb{F}_p((T)) \).
- Section 2 of these notes by Stevenhagen;
- these notes by Clark;
- these notes by Sutherland;
- these notes by Bouyer.
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