Math8410Ex2.4David

Math8410Ex2.4David - Extension of a valuation in an...

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Unformatted text preview: Extension of a valuation in an arbitrary ﬁeld extension David Krumm The purpose of this note is to prove the following result: Let (K,v) be a valued ﬁeld and L/K be an arbitrary extension. Then there is a valuation on L extending v . The idea of the proof is to use the known case of a ﬁnite extension, and to build on the results of exercises 2.2 and 2.3. The argument is a standard application of Zorn’s lemma. Let Σ be the collection of all valued ﬁelds (F, w) such that F is an intermediate extension K ⊆ F ⊆ L and w extends v . We deﬁne a partial order on Σ by (F1 , w1 ) ≤ (F2 , w2 ) if F1 ⊆ F2 and w2 extends w1 . Clearly this makes (Σ, ≤) into a nonempty partially ordered set. Suppose now that {(Fa , va )} is a totally ordered subset of Σ and let f = a Fa . Then K ⊆ f ⊆ L and by exercise 2.2 there is a valuation w on f extending each va . Since each va extends v , then w is an extension of v . Therefore, (f, w) ∈ Σ is an upper bound for the chain {(Fa , va )}. By Zorn’s lemma, there is a maximal element (F, W ) ∈ Σ. We claim that F = L. To see this, let t ∈ L. The valuation W can be extended to F (t) (by exercise 2.3 if t is transcendental over F , and by known results if t is algebraic), which implies that F (t) = F by maximality of (F, W ), and hence t ∈ F . This shows that F = L and we conclude that the valuation v can be extended to L. 1 ...
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