College Algebra Exam Review 314

College Algebra Exam Review 314 - 324 7. FIELD EXTENSIONS...

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 324 7. FIELD EXTENSIONS FIRST LOOK Since K L , there exists an element a 1 2 L n K . Then K K.a 1 / L , dim K .K.a 1 // > 1 , and dim K.a 1 / .L/ D dim K .L/= dim K .K.a 1 // < dim K .L/: By the induction hypothesis applied to the pair K.a 1 / L , there ex- ists a natural number n and there exist finitely many elements a 2 ;:::;a n , such that L D K.a 1 /.a 2 ;:::;a n / D K.a 1 ;a 2 ;:::;a n /: This completes the proof. n The set I of polynomials p.x/ 2 KOEx satisfying p./ D is the kernel of the homomorphism ' W KOEx ! L , so is an ideal in KOEx . We have I D f.x/KOEx , where f.x/ is an element of mini- mum degree in I , according to Proposition 6.2.27 . The polynomial f.x/ is necessarily irreducible; if it factored as f.x/ D f 1 .x/f 2 .x/ , where deg .f / > deg .f i / > 0 , then D f./ D f 1 ./f 2 ./ . But then one of the f i would have to be in I , while deg .f i / < deg .f / , a contra- diction. The generator f.x/ of I is unique up to multiplication by a...
View Full Document

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online