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Unformatted text preview: 422 9. FIELD EXTENSIONS – SECOND LOOK 9.2. Splitting Fields Now, we turn our point of view around regarding algebraic extensions. Given a polynomial f.x/ 2 KOExŁ , we produce extension fields K L in which f has a root, or in which f has a complete set of roots. Proposition 7.3.6 tells us that if f.x/ is irreducible in KOExŁ and ˛ is a root of f.x/ in some extension field, then the field generated by K and the root ˛ is isomorphic to KOExŁ=.f.x// ; but KOExŁ=.f.x// is a field, so we might as well choose our extension field to be KOExŁ=.f.x// ! What should ˛ be then? It will have to be the image in KOExŁ=.f.x// of x , namely, OExŁ D x C .f.x// . (We are using OEg.x/Ł to denote the image of g.x/ in KOExŁ=.f.x// .) Indeed, OExŁ is a root of f.x/ in KOExŁ=.f.x// , since f.OExŁ/ D OEf.x/Ł D in KOExŁ=.f.x// . Proposition 9.2.1. Let K be a field and let f.x/ be a monic irreducible element of KOExŁ . Then (a) There is an extension field L and an element ˛ 2 L such that f.˛/f....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08