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ma5c-hw1-soln

# ma5c-hw1-soln - M a 5c HOMEWORK 1 SOLUTION SPRING 09 The...

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Ma 5 c HOMEWORK 1 SOLUTION SPRING 09 The exercises are taken from the text, Abstract Algebra (third edi- tion) by Dummit and Foote, unless stated otherwise. Page 519, 4 . Deﬁne φ : Q [ 2] Q [ 2] via a + b 2 7→ a - b 2, where a,b Q . Then φ ( a + b 2) + φ ( c + d 2) = a - b 2 + c - d 2 = a + c - ( b + d ) 2 = φ (( a + c ) + ( b + d ) 2) = φ ( a + b 2 + c + d 2). φ ( a + b 2) φ ( c + d 2) = ( a - b 2)( c - d 2) = ( ac +2 bd ) - ( ad + bc ) 2 = φ ( ac + 2 bd + ( ad + bc ) 2) = φ (( a + b 2)( c + d 2)). Therefore φ is a homomorhpism. It’s obvious that φ is both injective and surjective. Page 530, 12 . We know that [ E : F ] divides [ K : F ] = p . Therefore [ E : F ] = 1, in which case E = F since E contains F , or [ E : F ] = p = [ K : F ], in which case E = K , since E is a subﬁeld of K . Page 530, 13 . Note that [ Q ( α 1 ) : Q ] | 2 since α 1 is a root of x 2 - α 2 1 Q [ x ]. Similarly, for any

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ma5c-hw1-soln - M a 5c HOMEWORK 1 SOLUTION SPRING 09 The...

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