ahw9 - Math 5126 Monday March 31 Ninth Homework Solutions...

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Math 5126 Monday, March 31 Ninth Homework Solutions 1. (a) Let ω be a primitive p th root of 1. Then the minimal polynomial of ω is x p - 1 + x p - 2 + ··· + x + 1; in particular it has degree p - 1, and the primitive p th roots of 1 are precisely ω r where 1 r p - 1. Suppose we have a linear relation a 1 ω + ··· + a p - 1 ω p - 1 where a i Q for all i . Since ω 6 = 0, we see that a 1 + a 2 ω + ··· + a p - 1 ω p - 2 = 0 . This shows that ω satisfies the polynomial a 1 + a 2 x + ··· + a p - 2 x p - 3 + a p - 1 x p - 2 , which has smaller degree than the minimal polynomial, so this polynomial must be the zero polynomial. Therefore all the a i are 0 and we conclude that the ω r are linearly independent. (b) Note that Φ n ( x ) and Φ n / p ( x p ) divide x n - 1, hence the roots of these two poly- nomials are distinct. Let α , β C . Then α satisfies Φ n ( x ) if and only if it is a primitive n th root of 1, and α satisfies Φ n / p ( x p ) if and only if β p is a primi- tive n / p th root of 1. Suppose β p is a primitive n / p th root of 1. Then certainly β n = 1, so β is an n th root of 1. Suppose m is a positive integer and β m = 1. Then
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This note was uploaded on 01/29/2009 for the course MATH 5125 taught by Professor Palinnell during the Fall '07 term at Virginia Tech.

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ahw9 - Math 5126 Monday March 31 Ninth Homework Solutions...

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