Exam3 - R itself. [20 pts] 6. Suppose R is a ring such that...

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MATH 373: Exam 3 Monday, April 26, 2010 Answer all questions in the answer booklet. Please do these in order. You are expected to justify your answers in a manner that an average MATH 373 student should be able to follow. This includes sentences for clarification. 1. Let ϕ : R S be a ring homomorphism which is onto and let A be an ideal of R . Prove that ϕ ( A ) is an ideal of S . [11 pts] 2. Prove or disprove: { (3 x, y ) | x, y Z } is a maximal ideal in Z × Z . [10 pts] 3. Describe the elements in Q ( 3 - 2). Justify carefully using an appropriate theorem. [10 pts] 4. Determine if the following polynomials are irreducible over Q . Justify your answer. [9 pts] (a) f ( x ) = 3 x 2 - 7 x - 5 (b) g ( x ) = x 5 - 4 x + 2 (c) h ( x ) = x 4 + x 2 + x + 1 Answer 3 of the following 5 questions, worth 20 points each. If you answer more than 3, either clearly indicate which ones you want graded or I will simply grade the first 3 that I see. 5. Let R be a commutative ring with unity. Prove R is a field IFF the only ideals are (0) and
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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 2-5). (b) Prove im = Q [ 5]. (c) Prove Q [ x ] / ( x 2-5) = 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.

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