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Suppose $p$ and $q$ primes and $p$ is odd. Then, are there nice and clever ways to factorize polynomials of the form $$f(x)=1+x+\cdots +x^{p-1}$$ in the ring $\mathbb{F}_q[x]$ ? What about the case when $q=2$ ?

I know that there are factorization algorithms but they are too general. I want to know if there are clever ways to do this for these special type of polynomials. Like for example, in $\mathbb{F}_2$ one might add terms $x^r+x^r$ and rearrange them so that $f(x)$ gets factorized.

In case there are no good methods to factorize, are there nice ways to check whether $f(x)$ is irreducible in $\mathbb{F}_q[x]$ ?

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Since $q^{p-1}-1$ is divisible by $p$, $x^{q^{p-1}}-x$ is divisible by $x^p-1=(x-1)(1+x+x^2+\dots+x^{p-1})$. In general, $x^{q^n}-x$ is divisible by exactly the prime polynomials of degree $d|n$. Therefore, the only prime factors of $1+x+\dots +x^{p-1}$ can be primes of degree $d|p-1$. Not sure where that leads you, however. –  Thomas Andrews Jul 1 '13 at 16:48
    
The general theory is described in Dtseng's (+1) answer. A similar approach will see you home for all exponents $p$ (even non-prime) as long as $\gcd(p,q)=1$. See this answer to a related question. In the case $p$ prime you get the extra bit that all the irreducible factors of $f(x)$ will have the same degree. A noteworthy observation is that $f(x)$ is irreducible, if and only if $q$ is a primitive element of $\mathbb{F}_p^*$. –  Jyrki Lahtonen Jul 2 '13 at 14:00
    
It just occured to me that the algorithm for finding the idempotents of cyclic codes of length $p$ with alphabet $\mathbb{F}_q$ can be tweaked to produce a factorization as well. It is easy to give a spanning set of idempotents of the ring $R=\mathbb{F}_q[x]/(x^p-1)$. Then multiplying them in $R$ allows us to find the basic idempotents. The basic idempotents $e(x)$ have the property that $(x^p-1)/\gcd(e(x),x^p-1)$ is an irreducible factor (possibly together with the trivial factor $x-1$, I'm not sure). The theory is described in J.H. van Lint's book from the 70s (earlier than his GTM book). –  Jyrki Lahtonen Jul 6 '13 at 18:32

1 Answer 1

We can write $f(x)=\frac{x^p-1}{x-1}$, so it suffices to see how $x^p-1$ factors.

If $p=q$, then $x^p-1=(x-1)^p$.

If $q\neq p$, then $x^p-1$ is separable (and its solutions are the $p^{th}$ roots of unity). We know what the multiplicative group $\mathbb{F}_{q'}^{\times}$ is for any field extension $\mathbb{F}_{q'}$ of $\mathbb{F}_{q}$ (namely, a cyclic group of order $q'-1$). Then we can solve for the minimum value of $q'$ such that $x^p-1$ splits by finding when $p$ divides $q'-1$. Then, to factor over $\mathbb{F}_{q}$, we can take Galois orbits of the $p^{th}$ roots of unity under the Galois extension $\mathbb{F}_{q'}$ of $\mathbb{F}_{q}$.

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