practice4b - MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS...

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MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS April 15, 2007 Here are a few practice problems on rings. Try to work these WITHOUT looking at the solutions! After you write your own solution, you can compare to my solution. Your solution does not need to be identical—there are often many ways to solve a problem—but it does need to be CORRECT. 1. Work out the rule of computation in the ring R [ x ] / ( f ), where f ( x ) = x 3 - 1. Note that the quotient ring consists of elements a + bx + cx 2 + ( f ). Solution For easier notation, let I = ( f ). Addition is easy: ( a 1 + b 1 x + c 1 x 2 + I ) + ( a 2 + b 2 x + c 2 x 2 + I ) = ( a 1 + a 2 ) + ( b 1 + b 2 ) x + ( c 1 + c 2 ) x 2 + I. Multiplication is harder. The key fact is that x 3 - 1 is in I , so x 3 - 1 + I = I . Therefore x 3 + I = 1 + I . Similarly, x 4 + I = xx 3 + I = ( x + I )( x 3 + I ) = ( x + I )( 1 + I ) = x + I. Hence, ( a 1 + b 1 x + c 1 x 2 + I )( a 2 + b 2 x + c 2 x 2 + I ) = a 1 a 2 + ( b 1 a 2 + a 1 b 2 ) x + ( c 1 a 2 + b 1 b 2 + a 1 c 2 ) x 2 + ( a 1 c 2 + c 1 a 2 ) x 3 + c 1 c 2 x 4 + I = a 1 a 2 + ( b 1 a 2 + a 1 b 2 ) x + ( c 1 a 2 + b 1 b 2 + a 1 c 2 ) x 2 + ( a 1 c 2 + c 1 a 2 ) + c 1 c 2 x + I = ( a 1 a 2 + a 1 c 2 + c 1 a 2 ) + ( b 1 a 2 + a 1 b 2 + c 1 c 2 ) x + ( c 1 a 2 + b 1 b 2 + a 1 c 2 ) x 2 + I. ± 2. Let F be a ±eld. Show that a cubic polynomial in F [ x ] either has a root in F or is irreducible over F . Solution
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This note was uploaded on 08/25/2011 for the course MATH 4107 taught by Professor Staff during the Spring '10 term at University of Florida.

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practice4b - MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS...

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