Math 254A. Absolute Values on Number Fields and Function Fields

This handout describes in more detail the situation with absolute values on number fields and function fields of dimension 1 over an arbitrary field.

Throughout this handout, all function fields are assumed to be of dimension 1 (i.e., transcendence degree 1 ) over a field of constants F .

We start by showing how the construction of the completion of a valued field behaves with respect to a field extension of valued fields.

Proposition 1. Let L/K be a finite extension of valued fields such that the absolute value on L extends the absolute value on K. Then there is a commutative diagram

L ⟶^{j} L̂
↑^{ϕ}  ↑^{ψ}
K ⟶^{i} K̂

in which ϕ is the inclusion map and i and j are the maps constructed in Theorem 5a in the handout “Valued rings and valued fields.” Moreover, L̂ = j(L)ψ(K̂) .

Proof. Let i, j, and ϕ be as given. Then the existence of ψ such that the diagram commutes follows from Theorem 5b of the earlier handout. We regard all of the fields in the above diagram as valued subfields of L̂, so that all of the maps in the diagram are inclusion maps.

Now regard L as a vector space over K, and fix a basis. By II Prop. 4.9, a sequence in L is Cauchy (in the absolute value on L) if and only if all of the coordinate sequences with respect to this basis are Cauchy, and therefore this basis spans L̂ as a vector space over K̂. This proves the last assertion. □

The following result was only partially proved in class (on 3 November).

Proposition 2. Let K be a field. Then the map

σ ∈ Hom(K, ℂ) ↦ (|x| = |σ(x)| ∀ x ∈ K)  (2.1)

determines a well-defined bijection from Hom(K, ℂ)/(complex conjugation) to the set of archimedean places of K.

Proof. Clearly a map σ: K → ℂ and its complex conjugate determine the same absolute value on K, and this absolute value is archimedean, so the map (2.1) is well defined. It is surjective by a corollary of Ostrowski’s theorem from class (November 3).

It remains only to show that the map (2.1) is injective.

Let σ_{1}, σ_{2} ∈ Hom(K, ℂ) and assume that they are mapped to the same place of K. It was proved in class on November 1 that if two absolute values | · |_{1} and | · |_{2} on a field are equivalent, then | · |_{1} = | · |_{2}^{s} for some (fixed) s > 0 . Therefore we have |σ_{1}(x)| = |σ_{2}(x)|^{s} for all x ∈ K. However, comparing the values at x = 2 gives that s = 1 , so |σ_{1}(x)| = |σ_{2}(x)| for all x ∈ K.

Now for any given x ∈ K, we have |σ_{1}(x)| = |σ_{2}(x)| and |σ_{1}(x−1)| = |σ_{2}(x−1)|. Therefore, comparing the triangle with vertices 0, 1, σ_{1}(x) ∈ ℂ with the triangle with vertices 0, 1, σ_{2}(x) ∈ ℂ, we find that both triangles have the same side lengths, so either σ_{1}(x) = σ_{2}(x) or σ_{1}(x) = \overline{σ_{2}(x)} .

Now if σ_{1}(K) ⊆ ℝ, then by the above we also have σ_{2}(K) ⊆ ℝ, and therefore σ_{1} = σ_{2} .

Hence we may assume that σ_{1}(K) ⊈ ℝ. Fix x ∈ K such that σ_{1}(x) ∉ ℝ. After replacing σ_{2} with its complex conjugate if necessary, we may assume that σ_{2}(x) = σ_{1}(x) . Then we are done if σ_{1} = σ_{2} . If not, then there is some y ∈ K such that σ_{1}(y) ≠ σ_{2}(y) . Then we must have σ_{1}(y) = \overline{σ_{2}(y)} . After replacing y with 1/y if necessary, we may assume that the imaginary parts of σ_{1}(x) and σ_{1}(y) have the same sign. Then the imaginary parts of σ_{2}(x) = σ_{1}(x) and σ_{2}(y) = \overline{σ_{1}(y)} have different signs, so

|σ_{1}(x − y)| = |σ_{1}(x) − σ_{1}(y)| < |σ_{1}(x) − \overline{σ_{1}(y)}| = |σ_{2}(x) − σ_{2}(y)| = |σ_{2}(x − y)| ,

a contradiction. Therefore we must have σ_{1} = σ_{2} . □

Corollary 3. Let K be a number field, and let ρ_{1}, …, ρ_{r}, σ_{1}, \overline{σ}_{1}, …, σ_{s}, \overline{σ}_{s} be the distinct embeddings of K into ℂ. Then the distinct archimedean places of K are represented by the pull-backs of the standard absolute value on ℂ (or ℝ) via ρ_{1}, …, ρ_{r}, σ_{1}, …, σ_{s}.

Proof. Immediate. □

Therefore, if L/K is an extension of number fields, then for any archimedean place v of K, the set of places of L lying over v is nonempty and finite. Moreover, by Corollary 8 in the earlier handout, all of these places (of L) are archimedean.

If K is a function field with constant field F , then there are no archimedean absolute values on K that restrict to the trivial absolute value on F , again by Corollary 8 of the earlier handout (using the fact that the trivial absolute value is non-archimedean).

This says all that we intend to say in this handout about archimedean places, so we now concentrate on non-archimedean places.

Lemma 4. Let L/K be an algebraic field extension. Then the only absolute value on L that extends the trivial absolute value on K is the trivial absolute value on L. (In other words, Theorem II 4.8 also holds if the absolute value on K is trivial; note that every field K is complete with respect to the trivial absolute value.)

Proof. Let | · | be an absolute value on L that extends the trivial absolute value on K. We shall show that | · | is also trivial on L.

Since the trivial absolute value on K is non-archimedean, the absolute value on L is also non-archimedean (by Corollary 8 of the earlier handout).

We now claim that |α| ≤ 1 for all α ∈ L. Indeed, let α ∈ L and write

Irr_{α,K}(X) = X^{n} + a_{n−1}X^{n−1} + ⋯ + a_{0} .

Then

\begin{aligned}|α|^{n} & = | − a_{n−1}α^{n−1} − a_{n−2}α^{n−2} − ⋯ − a_{0}| \\ & ≤ max\{|a_{n−1}||α|^{n−1}, |a_{n−2}||α|^{n−2}, …, |a_{0}|\} \end{aligned}

≤ max{|α|^{n−1}, 1} ,

which gives |α| ≤ 1 . But also, if α ≠ 0 then |1/α| ≤ 1 , so we must have |α| = 1 . Thus the absolute value on L must be trivial. □

We now consider the following result for extensions of valuations coming from nonzero primes of Dedekind rings.

Lemma 5. Let L/K be a finite extension of number fields or of function fields over a constant field F . If K and L are number fields, then let A = O_{K} and B = O_{L}; if they are function fields, then let A be a Dedekind ring of finite type over F whose fraction field is K, and let B be the integral closure of A in L. Note that in either case B is finite over A. Let | · | be an absolute value on L. Assume that there exist a nonzero prime 𝔭 of A and a constant C > 1 such that |x| = C^{−ν_{𝔭}(x)} for all x ∈ K^{∗}. Then there is a prime ideal 𝔮 of B lying over 𝔭 such that |x| = C^{−ν_{𝔮}(x)/e} for all x ∈ L^{∗}, where e = e_{𝔮/K} is the ramification index.

Proof. This follows fairly easily from Exercise 6 on page 166 (which may appear on a future homework assignment). Here we give a different proof, which applies only to this case.

First consider the special case in which L/K is normal.

Let w: L^{∗} → ℝ be the valuation on L defined by w(x) = − log |x|, and let v = w|_{K} be its restriction to K. Note that v(x) = ν_{𝔭}(x) log C for all x ∈ K^{∗}.

Fix an algebraic closure \overline{L}_{w} of L_{w} (hence of K_{v}).

Choose a prime 𝔮_{0} of B lying over 𝔭, and let w_{0}(x) = (ν_{𝔮_{0}}(x)/e_{𝔮_{0}/K}) log C for all x ∈ L^{∗}. This is an extension of v to L, so by II Thm. 8.1 (i), there is an embedding τ: L → \overline{L}_{w} over K such that w_{0} = w ∘ τ .

Since L is normal over K, the image of τ equals the image of the canonical injection of L into L_{w}, so τ is an element of Gal(L/K) . Let 𝔮 = τ(𝔮_{0}) . Since τ is an automorphism of L over K, we have e_{𝔮_{0}/K} = e_{𝔮/K} = e, and therefore

− log |x| = w(x) = w_{0}(τ^{−1}(x)) = \frac{ν_{𝔮_{0}}(τ^{−1}(x))}{e} log C = \frac{ν_{τ(𝔮_{0})}(x)}{e} log C = \frac{ν_{𝔮}(x)}{e} log C

for all x ∈ L^{∗}, as was to be shown.

For the general case, let L′ be a finite normal extension of K containing L, choose an extension of | · | to L′, let B′ be the integral closure of B in L′, and let 𝔮′ be a nonzero prime of B′ that satisfies the conditions of the lemma for L′/K. Then the lemma for L/K follows by letting 𝔮 = 𝔮′ ∩ L and applying multiplicativity of the ramification index in towers. □

We are now able to prove the following generalization of II Prop. 3.7. (This result is somewhat academic, however, since we usually take the set of places of a number field to be a given.)

Theorem 6. Let K be a number field, and let | · | be a nontrivial non-archimedean absolute value on K. Then there exist a prime 𝔭 of K and a constant C > 1 such that |x| = C^{−v_{𝔭}(x)} for all x ∈ K^{∗}. In particular, this defines a canonical bijection between the set of non-archimedean places of K and the set of primes of K.

Proof. By Lemma 4, the restriction of | · | to ℚ is nontrivial, so by II Prop. 3.7 it is equivalent to | · |_{p} for some rational prime p. By Lemma 5, there exist a prime ideal 𝔭 of O_{K} lying over p and C > 1 that satisfy the condition of the theorem. Clearly any two distinct primes 𝔭 of K give inequivalent absolute values, hence distinct places, and this gives the bijection. □

A similar result holds in the function field case, but proving it requires a result similar to II Prop. 3.7.

Lemma 7. Let F be an arbitrary field, let K = F (t) (with t an indeterminate), and let | · | be a nontrivial non-archimedean absolute value on K whose restriction to F is trivial. Then there exist a prime 𝔭 of either F [t] or F [1/t] (or both) and a constant C > 1 such that |x| = C^{−v_{𝔭}(x)} for all x ∈ K^{∗}.

Proof. After replacing t with 1/t if necessary, we may assume that |t| ≤ 1 . Let A = F [t] . Then |x| ≤ 1 for all x ∈ A. Let 𝔭 = {x ∈ A : |x| < 1}. Then (since | · | is non-archimedean) 𝔭 is a prime ideal in A. Also 𝔭 ≠ (0) (because | · | is nontrivial and A generates K). Therefore 𝔭 = (π) for some irreducible π ∈ F [t] . We then have |x| = |π|^{v_{𝔭}(x)} for all nonzero x ∈ A, and therefore the same equation holds for all x ∈ K^{∗}. □

Theorem 8. Let K be a function field with constant field F , and let t ∈ K be transcendental over F (so that K is finite over F (t) ). Let | · | be a nontrivial absolute value on K which is trivial on F . Then there exist (i) a nonzero prime ideal 𝔭 in the integral closure of either F [t] or F [1/t] in K and (ii) a constant C > 1 , such that |x| = C^{−v_{𝔭}(x)} for all x ∈ K^{∗}. In particular, this defines a canonical bijection between the set of places of K trivial on F and the set of closed points on the nonsingular projective curve over F with function field K.

Proof. By Lemma 4, the restriction of | · | to F (t) is nontrivial, so by Lemma 7 there exist a prime ideal 𝔭_{0} of A, with A equal to either F [t] or F [1/t] , and C_{0} > 1 such that |x| = C_{0}^{−v_{𝔭_{0}} (x)} for all x ∈ F (t)^{∗}. Let B be the integral closure of A in K. Then, by Lemma 5, there is a prime ideal 𝔭 of B such that |x| = C^{−v_{𝔭}(x)} for all x ∈ K^{∗}, where C = C_{0}^{1/e} and e = e_{𝔭/𝔭_{0}} is the ramification index. Now (please forgive the algebraic geometry) Spec B is an open affine subset of the curve mentioned in the statement of the theorem, and 𝔭 corresponds to a closed point of Spec B. With some additional work in algebraic geometry, we obtain the required bijection. □