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The excercise is to factorize $\Phi_7$ into irreducible factors modulo $2$.

A theorem in my course: Let $\mathbb{F}_q$ be a finite field of characteristic $p$ and $n\in\mathbb{N}$ such that $\gcd(q,n)=1$. Then the degree of every irreducible factor of $\Phi_n$ (mod $p$) in $\mathbb{F}_q$ is equal to the order of $q$ in $(\mathbb{Z}/n\mathbb{Z})^*$.

I used this and concluded that $2\rightarrow 4 \rightarrow 8 = 1$ (mod $7$), so all the irreducible factors are of degree 3. Now the actual question. Given this knowledge how do I find the factorization? By trial and error I found $$ X^6 + X^5 + X^4 + X^3 + X^2 + X + 1 = (X^3+X^2+1)(X^3+X+1) \text{ (mod $2$)}. $$
Can someone help me to find this actual factorization systematically? That is for arbitrary $n$ and $q$, in which case trial and error won't work. Thank you!

Edit: Maybe my question is too greedy. If somebody could provide me with some insight on how to solve the factorization for very small numbers, that would already be a tremendous help.

Marc
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    Small numbers you say? A related question has an example, but that is largely ad hoc. May be Berlekamp's factorization algorithm helps you in some other cases? An even smaller example is studied here. Note that unless you know the low degree irreducible polynomials over $\Bbb{F}_2$ by heart, then the trick of going reciprocal helps in your case. Sorry that this is mostly just a bag of tricks. – Jyrki Lahtonen Mar 23 '14 at 20:54
  • Thank you very much! Especially the first example using the frobenuis homomorphism is very useful. – Marc Mar 23 '14 at 20:59

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