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College Algebra Exam Review 314

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

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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 KOExŁ satisfying p.˛/ D 0 is the kernel of the homomorphism ' ˛ W KOExŁ ! L , so is an ideal in KOExŁ . We have I ˛ D f .x/KOExŁ , 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 0 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 nonzero element of K , so there is a unique monic polynomial f .x/ such that I ˛ D f .x/KOExŁ , called the
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