A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

Chapter 19
Extensions and Normalizations
of Valuations
19.1 Extensions of Valuations
In this section we continue to tacitly assume that all valuations are nontrivial. We
do not assume all our valuations satisfy the triangle
Suppose K ⊂ L is a finite extension of fields, and that | · | and · are valuations
on K and L, respectively.
Definition 19.1.1 (Extends). We say that · extends | · | if |a| = a for all
a ∈ K.
Theorem 19.1.2. Suppose that K is a field that is complete with respect to | · | and
that L is a finite extension of K of degree N = [L : K]. Then there is precisely one
extension of | · | to K, namely
a = Norm L/K(a) 1/N , (19.1.1)
where the Nth root is the non-negative real Nth root of the nonnegative real number
Norm L/K(a) .
Proof. We may assume that | · | is normalized so as to satisfy the triangle inequality.
Otherwise, normalize | · | so that it does, prove the theorem for the normalized
valuation | · | c , then raise both sides of (19.1.1) to the power 1/c. In the uniqueness
proof, by the same argument we may assume that · also satisfies the triangle
inequality.
Uniqueness. View L as a finite-dimensional vector space over K. Then · is a
norm in the sense defined earlier (Definition 18.1.1). Hence any two extensions · 1 and · 2 of | · | are equivalent as norms, so induce the same topology on K. But as
we have seen (Proposition 16.1.4), two valuations which induce the same topology
are equivalent valuations, i.e., · 1 = · c
2 , for some positive real c. Finally c = 1
since a1 = |a| = a2 for all a ∈ K.
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