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单词 FieldHomomorphismsFixPrimeSubfields
释义

field homomorphisms fix prime subfields


Theorem.

Let F and K be fields having the same prime subfieldMathworldPlanetmath L and φ:FK be a field homomorphism. Then φ fixes L.

Proof.

Without loss of generality, it will be assumed that L is either or /c.

Since φ is a field homomorphism, φ(0)=0, φ(1)=1, and, for every xF, φ(-x)=-φ(x).

Let n and c be the characteristicPlanetmathPlanetmath of F. Then

This the proof in the case that c is prime.

Now consider c=0. Let x. Then there exist a,b with b>0 such that x=ab. Thus, bφ(x)=j=1bφ(ab)=φ(j=1bab)=φ(a)=a. Therefore, φ(x)=ab=x. Hence, φ fixes .∎

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更新时间:2025/5/4 11:44:17