Re: Discriminant

trusiki@xxxxxxxxx wrote:

Prove that the discriminant of the cyclotomic polynomial Phi_p(x) of
the p-th roots of unity for an odd prime p is (-1)^((p-1)/2)*p^(p-2).

Let s1,..,s_p-1 be the p-th roots of unity different from 1.

The discriminant is by definition the square of Prod_i<j{ ( s_i - s_j) }

Now take the product Prod_i<>j { s_i - s_j ). What is this product in
respect to the previous?

Now in write each factor (si-sj) as si(1-sj/si) [not that sj/si equals sk
for some k). Now the previous product becomes s1^e1*s2^e2...sp^ep*
Prod_k{1-s^k}^fk. Try fo find out the exponents e1, ... ep and f1,... fk.

What does the last factor in this product remind you of if all the exponents
f_k were even? Hint: Think of the decomposition of Phi_p and how it splits
in linear factors. How many terms does this polynomial have?
.