Let $\alpha$ be an $n$ th primitive root of unity in $GF(q^N)$, where $q$ is a power of a prime, $N$ is the order of $q$ in $\mathbb{Z_n^{*}}$ and $n$ is a product of two primes. Then why $x^n -1 =\prod_{i=0}^{n-1}(x-\alpha^i)$?
Primitive root of unity
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abstract-algebra
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0This should only be true if $n$ is coprime to $|q^N|$. – 2012-11-28
1 Answers
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Each side of the equation is a monic polynomial in $x$ of degree $n-1$, and they agree at the $n$ points $\alpha^i$, $0\le i\le n-1$. That should do it.