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Unformatted text preview: R itself. [20 pts] 6. Suppose R is a ring such that x 2 = x for all x R . Prove that R is commutative. [20 pts] 7. (Chinese Remainder Theorem) Let F be a eld and f ( x ) , f ( x ) F [ x ] be relatively prime. Given polynomials b ( x ) , b ( x ) F [ x ] show that there exists a polynomial c ( x ) F [ x ] such that c ( x ) + ( f ( x )) = b ( x ) + ( f ( x )) and c ( x ) + ( f ( x )) = b ( x ) + ( f ( x )). [20 pts] 8. Consider x 2 + x + 1 Z 5 [ x ] [20 pts] (a) Prove f ( x ) is irreducible over Z 5 . (b) In which eld extension does it have a zero? (c) Factor it in this eld extension. 9. Dene : Q [ x ] R by ( f ( x )) = f ( 5). You can assume is a ring homomorphism [20 pts] (a) Prove ker = ( x 25). (b) Prove im = Q [ 5]. (c) Prove Q [ x ] / ( x 25) = Q [ 5]...
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This note was uploaded on 04/19/2011 for the course MATH 21373 taught by Professor Johnson during the Spring '11 term at Carnegie Mellon.
 Spring '11
 Johnson
 Math, Algebra

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