Unformatted text preview: 9.2. SPLITTING FIELDS 423 Proof. The ring isomorphism W KOExŁ ! K OExŁ induces a field isomor- phism Q W KOExŁ=.f / ! K OExŁ=. .f // , satisfying Q .k C .f // D .k/ C . .f // for k 2 K . Now, use K.˛/ Š KOExŁ=.f / Š K OExŁ=. .f // Š K .˛ / . n Now, consider a monic polynomial f.x/ 2 KOExŁ of degree d > 1 , not necessarily irreducible. Factor f.x/ into irreducible factors. If any of these factors have degree greater than 1, then choose such a factor and adjoin a root ˛ 1 of this factor to K , as previously. Now, regard f.x/ as a polynomial over the field K.˛ 1 / , and write it as a product of irreducible factors in K.˛ 1 /OExŁ . If any of these factors has degree greater than 1, then choose such a factor and adjoin a root ˛ 2 of this factor to K.˛ 1 / . After repeating this procedure at most d times, we obtain a field in which f.x/ factors into linear factors. Of course, a proper proof goes by induction; see Exercise 9.2.1 ....
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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