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ma5c-hw2-soln

# ma5c-hw2-soln - M a 5c HOMEWORK 2 SOLUTION SPRING 09 The...

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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 α K . Let β be any other root of f . By Theorem 8, there is an isomorphism σ : F ( α ) F ( β ). Now K = K ( α ) is still the splitting field of { f i ( x ) } over F ( α ). Also, K ( β ) is the splitting field of { f i ( x ) } over F ( β ). By Theorem 27, σ extends to an isomorphism φ : K K ( β ). In particular, [ K : F ] = [ K ( β ) : F ], and K = K ( β ). 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 ( α 1 , α 2 , . . . , α n ). Let f i be the irreducible polynomial over F with α i as a root. Then by assumption, f i splits completely in K [ x ], and K contains the splitting field of { f i } . But any subfield of K would not contain all the roots of. Therefore K is the splitting field of { f i } .

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

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