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Unformatted text preview: R } is a commutative ring under pointwise operations. (i) If R is a commutative ring, prove that : R [ x ] F ( R ) , given by : f ( x ) 7 f [ , is a homomorphism. Solution. Absent. (ii) If k is an in f nite f eld, prove that is an injection. Solution. Absent. 3.58 Find the gcd of x 2 x 2 and x 3 7 x + 6 in F 5 [ x ] , and express it as a linear combination of them. Solution. The Euclidean algorithm shows that gcd ( x 2 x 2 , x 3 7 x + 6 ) = x 2 , and x 2 = 1 4 ( x 3 7 x + 6 ) + 1 4 ( x + 1 )( x 2 x 2 )....
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 Fall '11
 KeithCornell

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