ahw10 - Math 5125 Monday, November 14 Tenth Homework...

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Math 5125 Monday, November 14 Tenth Homework Solutions 1. (a) Both sides have degree n and both sides have roots { ζ i | i = 1 , . . . , n } . Since both sides have constant term, it follows that both sides are equal. (b) More generally in C [ x , y ] , we have 1 - ( x / y ) n = n i = 1 ( 1 - ζ i x / y ) and we see that y n - x n = n i = 1 ( y - ζ i x ) . Since f n = h n - g n , we deduce that f n = n i = 1 ( h - ζ i g ) . Suppose a prime k C [ x ] divides two of the factors, say ( h - ζ i g ) , ( h - ζ j g ) where 1 i , j n , i 6 = j (so ζ i 6 = ζ j ). Then k divides ( ζ i - ζ j ) g and we deduce that k divides g , h , contrary to the hypothesis that g , h are relatively prime. (c) This follows from the fact that C [ x ] is a UFD. Indeed suppose f = k a 1 1 . . . k a d d , where the k i are distinct primes and a i 1. Then for each i , k i divides precisely one of the ( h - ζ j ) and none of the others, by relative primeness. We deduce that k na i i divides ( h - ζ j ) and the result follows. (d) By (c), we may write h - g = a n , h - ζ g = b n , h - ζ 2 g = c n , where a , b , c C [ x ] (we use n 3 here). Since h - g , h - ζ g , h - ζ 2 g are linearly dependent over
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ahw10 - Math 5125 Monday, November 14 Tenth Homework...

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