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Math Help - quad. residues

  1. #1
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    cycloatomic poly.

    See attachment. Thanks.
    Last edited by bigb; October 26th 2008 at 06:45 PM.
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  2. #2
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    Quote Originally Posted by bigb View Post

    Suppose that n is a positive integer, and m is the largest odd divisor of n. Show that x^n + 1 factors as a product of m irreducible polynomials.

    Hint: x^n + 1 = \frac{x^{2n}-1}{x^n - 1}.
    this is false! the correct statement is: x^n+1 factors as a product of \tau(m) irreducible polynomials, where \tau(m) is the number of divisors of m. here is why: so we have n=2^km, for some

    integer k \geq 0. see that A=\{2^{k+1}d: \ d \mid m \} is exactly the set of those divisors of 2n which do not divide n. obviously: |A|=\tau(m). now let \Phi_d(x) be the d-th cyclotomic polynomial. then:

    x^n + 1 = \frac{x^{2n} - 1}{x^n - 1} = \frac{\prod_{d \mid 2n} \Phi_d(x)}{\prod_{d \mid n}\Phi_d(x)}=\prod_{r \in A} \Phi_r(x). \ \ \Box
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