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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...
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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