College Algebra Exam Review 423

College Algebra Exam Review 423 - 9.4.3 ). By the induction...

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9.4. SPLITTING FIELDS AND AUTOMORPHISMS 433 Corollary 9.4.15. If K ± L is a finite–dimensional Galois extension and K ± M ± L is an intermediate field, then M ± L is a Galois extension. Proof. L is the splitting field of a separable polynomial over K , and, there- fore, also over M . n Proposition 9.4.16. If K ± L is a finite–dimensional Galois extension, then dim K L D j Aut K .L/ j : (9.4.3) Proof. The result is evident if K D L . Assume inductively that if K ± M ± L is an intermediate field and dim M L < dim K L , then dim M L D j Aut M .L/ j . Let ˛ 2 L n K and let p.x/ 2 KŒxŁ be the minimal polynomial of ˛ over K . Since L is Galois over K , p is separable and splits over L , by Theorem 9.4.14 . If ' 2 Iso K .K.˛/;L/ , then '.˛/ is a root of p , and ' is determined by '.˛/ . Therefore, deg .p/ D j Iso K .K.˛/;L/ j D Œ Aut K .L/ W Aut K.˛/ .L/Ł; (9.4.4) where the last equality comes from Equation (
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Unformatted text preview: 9.4.3 ). By the induction hy-pothesis applied to K./ , j Aut K./ .L/ j D dim K./ L is nite. Therefore, Aut K .L/ is also nite, and j Aut K .L/ j D deg .p/ j Aut K./ .L/ j D dim K .K.// dim K./ .L/ D dim K L; where the rst equality comes from Equation ( 9.4.4 ), the second from the induction hypothesis and the irreducibility of p , and the nal equality from the multiplicativity of dimensions, Proposition 7.3.1 . n Corollary 9.4.17. Let K L be a nitedimensional Galois extension and M an intermediate eld. Then j Iso K .M;L/ j D dim K M: (9.4.5) Proof. j Iso K .M;L/ j D Aut K .L/ W Aut M .L/ D dim K .L/ dim M .L/ D dim K M;...
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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.

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