hw10 - C x(d Write h-g = a n h g = b n h 2 g = c n where a...

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Math 5125 Friday, November 4 Tenth Homework Due 9:05 a.m., Friday November 11 1. Prove Fermat’s last theorem for C [ x ] , speciﬁcally prove that there is no solution to f n + g n = h n for n Z , n 3, with f , g , h C [ x ] , relatively prime (i.e. no prime divides two of them, and hence all three of them), and f , g , h all have degree at least 1. So assume by way of contradiction we have such a solution with deg f + deg g + deg h as small as possible. (a) Let ζ = e 2 π i / n . Prove that 1 - x n = n i = 1 ( 1 - ζ i x ) . (b) Prove that f n = n i = 1 ( h - ζ i g ) and that the factors ( h - ζ i g ) are relatively prime. (c) Prove that each factor ( h - ζ i g ) is an n th power in
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Unformatted text preview: C [ x ] . (d) Write h-g = a n , h- g = b n , h- 2 g = c n where a , b , c C [ x ] . Show that ( a ) n +( b ) n = ( c ) n for suitable nonzero complex numbers , , . (e) Complete the proof. (5 points) 2. Section 10.1, Exercise 8 on page 344. (3 points) 3. Section 10.1, Exercise 15 on page 344. (2 points) 4. Section 10.2, Exercise 3 on page 350. (2 points) 5. Section 10.2, Exercise 9 on page 350. (3 points) (5 problems, 15 points altogether)...
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