College Algebra Exam Review 414

College Algebra Exam Review 414 - and observe that ².p 1.x...

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424 9. FIELD EXTENSIONS – SECOND LOOK L ± p p p p p p p p p p p p p p p p p q q q q q Q L ± q q q q q ± q q q q q K;p.x/ ² q q q q q Q K; Q p.x/ Proof. The idea is to use Proposition 9.2.2 and induction on dim K .L/ . If dim K .L/ D 1 , there is nothing to do: The polynomial p.x/ factors into linear factors over K , so also Q p.x/ factors into linear factors over Q K , K D L , Q K D Q L , and ² is the required isomorphism. We make the following induction assumption: Suppose K ± M ± L and Q K ± Q M ± Q L are intermediate field extensions, Q ² W M ! Q M is a field isomorphism extending ² , and dim M L < n D dim K .L/ . Then there is a field isomorphism ± W L ! Q L such that ±.m/ D Q ².m/ for all m 2 M . Now, since dim K .L/ D n > 1 , at least one of the irreducible factors of p.x/ in KŒxŁ has degree greater than 1. Choose such an irreducible factor p 1 .x/
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Unformatted text preview: , and observe that ².p 1 .x// is an irreducible factor of Q p.x/ . Let ˛ 2 L and Q ˛ 2 Q L be roots of p 1 .x/ and ².p 1 .x// , respectively. By Proposition 9.2.2 , there is an isomorphism Q ² W K.˛/ ! Q K. Q ˛/ taking ˛ to Q ˛ and extending ² . L ± p p p p p p p p p p p p p p p p p q q q q q Q L ± q q q q q ± q q q q q K.˛/ Q ² q q q q q Q K. Q ˛/ ± q q q q q ± q q q q q K;p.x/ ² q q q q q Q K; Q p.x/ Figure 9.2.1. Stepwise extension of isomorphism. Now, the result follows by applying the induction hypothesis with M D K.˛/ and Q M D Q K. Q ˛/ . See Figure 9.2.1 n...
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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.

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