ma5c-hw2-soln

ma5c-hw2-soln - Ma 5 c HOMEWORK 2 SOLUTION SPRING 09 The...

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Unformatted text preview: Ma 5 c HOMEWORK 2 SOLUTION SPRING 09 The exercises are taken from the text, Abstract Algebra (third edi- tion) by Dummit and Foote. Page 545, 5 . Suppose K is a splitting field over F for a collection of { f i ( x ) } . Let f be an irreducible polynomial over F with a root K . Let be any other root of f . By Theorem 8, there is an isomorphism : F ( ) F ( ). Now K = K ( ) is still the splitting field of { f i ( x ) } over F ( ). Also, K ( ) is the splitting field of { f i ( x ) } over F ( ). By Theorem 27, extends to an isomorphism : K K ( ). In particular, [ K : F ] = [ K ( ) : F ], and K = K ( ). Therefore f splits completely in K [ x ]. Conversely, suppose any irreducible polynomial in F [ x ] that has a root in K splits completely in K [ x ]. Since K is a finite extension of F , K = F ( 1 , 2 ,..., n ). Let f i be the irreducible polynomial over F with i as a root. Then by assumption, f i splits completely in K [ x ], and...
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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-hw2-soln - Ma 5 c HOMEWORK 2 SOLUTION SPRING 09 The...

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