Unformatted text preview: 9.4.3 ). By the induction hypothesis applied to K.˛/ , j Aut K.˛/ .L/ j D dim K.˛/ L is ﬁnite. Therefore, Aut K .L/ is also ﬁnite, 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 ﬁrst equality comes from Equation ( 9.4.4 ), the second from the induction hypothesis and the irreducibility of p , and the ﬁnal equality from the multiplicativity of dimensions, Proposition 7.3.1 . n Corollary 9.4.17. Let K ± L be a ﬁnite–dimensional Galois extension and M an intermediate ﬁeld. Then j Iso K .M;L/ j D dim K M: (9.4.5) Proof. j Iso K .M;L/ j D Œ Aut K .L/ W Aut M .L/Ł D dim K .L/ dim M .L/ D dim K M;...
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 Fall '08
 EVERAGE
 Logic, Algebra, Mathematical Induction, Mathematical logic, Galois theory, Mathematical proof

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