{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

practice4b

# practice4b - MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

practice4b - MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online