College Algebra Exam Review 414

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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/
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , and observe that .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 .p 1 .x// , respectively. By Proposition 9.2.2 , there is an isomorphism Q W K./ ! Q K. Q / taking to Q and extending . 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 q q q q q Q K. Q / q q q q q q q q q q K;p.x/ 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...
View Full Document

Ask a homework question - tutors are online