ma5c-hw7-soln

ma5c-hw7-soln - M a 5c HOMEWORK 7 SOLUTION SPRING 09 The...

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Ma 5 c HOMEWORK 7 SOLUTION SPRING 09 The exercises are taken from the text, Abstract Algebra (third edi- tion) by Dummit and Foote, unless stated otherwise. Page 582, 17 . c) Gal ( K/F ) = { 1 } , with σ ( a + b D ) = a - b D . Therefore N K/F ( a + b D ) = ( a + b D )( a - b D ) = a 2 - b 2 D . d) Since [ F ( α ) : F ] = d , d | n follows from the tower law. α has d conjugates since the minimal polynomial is of degree d . Suppose H 0 is the subgroup of G corresponding to F ( α ), then H H 0 of index k = n d . Therefore, α is fixed by k embeddings of K , and each conjugate α i appears k times in Q σ ( α ). Furthermore, Q α i = ( - 1) d a o . Therefore N K/F ( α ) = Q σ ( α i ) = Q ( α i ) k = ( - 1) n ( a 0 ) k . 2.a) The minimal polynomial for 1 - ζ is Φ p r (1 - x ) = (1 - x ) p r - 1 (1 - x ) p r - 1 - 1 = 1 + t + ··· + t p - 1 , where t = (1 - x ) p r - 1 . N (1 - ζ ) = Φ p r (1 - 0) = p . b)Suppose
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This note was uploaded on 07/19/2010 for the course MA 5c taught by Professor Susamaagarwala during the Spring '09 term at Caltech.

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ma5c-hw7-soln - M a 5c HOMEWORK 7 SOLUTION SPRING 09 The...

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