Suppose F is a field and that ## f(x) ## is a non-constant polynomial in ##F[x]##. Since ##F[x] ## is a unique factorization domain, ## f(x) ## has an irreducible factor, ## p(x) ##. Then the fundamental theorem of field theory says that the field ## E = F[x]/<p(x)> ## contains a zero of ## f(x) ##. I am confused by the last statement.(adsbygoogle = window.adsbygoogle || []).push({});

## f(x) ## is an element of ## F[x] ##, not ## E ##, so what does it mean to say that E contains a zero of ## f(x) ##? For an element ## \alpha ## to be a zero of ## f(x) ##, it must be the case that ## \alpha \in F ## and ## f(\alpha) = 0 ## where ## 0 ## is the identity of F.

I'm a bit confused here. Perhaps someone could lend me a hand? Thanks!

BiP

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# Fundamental Theorem of Field extensions

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