hw12 - F 2 [ x ] / ( x 4 + x +1) is such a field and α =...

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MATH 3360 HOMEWORK SOLUTION 12 (Some computational problems are omitted.) p50, 3.5 To prove that h a i is an ideal, we need to show that if x ∈ h a i and s R , then xs ∈ h a i . Now by definition x = ra for some r , hence xs = ( rs ) a , which by definition is in h a i . p50, 3.7 Let I be an ideal in Z and let a be the least positive number in I . We show that I = h a i . First, it is easy to see that every multiple of a is in I , since addition is closed in the ideal. Conversely, suppose b I is not a multiple of a , then b = pa + c where | c | < a . Then c I and - c I . This is impossible since we assumed that a is the least positive number in I . p58, 4.5 The key is to find a field of 16 elements and find a primitive element of it. For example,
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Unformatted text preview: F 2 [ x ] / ( x 4 + x +1) is such a field and α = [ x ] is a primitive element. Then g ( x ) = ( x-α )( x-α 2 )( x-α 3 )( x-α 4 ). You need to simplify g ( x ) by the relation that α 4 + α + 1 = 0. (You don’t have to do simplifications in 4.1 because everyone knows how to multiply numbers and modulo them by 11. But here it is not directly known how to multiply these powers of α to get something in F 16 .) 1...
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This note was uploaded on 09/05/2011 for the course MATH 3360 taught by Professor Billera during the Spring '08 term at Cornell.

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