I need to reduce as much as possible the polynomial: $x^6+3x^5+2x^4+6x^3+4x^2+5x+2$ in the finite field $\Bbb F_7$. It has no roots over the field, and I don't think that it is necessary to check factorization with all the irreducible polynomials with deg 2 or 3. It was asked after the very first introduction of fields extensions. I tried (with no luck) to prove it's irreducible by extending the field where the polynomial has a root, tried by contradiction with dimensions, tried to mess with the roots and its squares multiplication groups that they generate and now I can't find how it helps me. I'll be glad for a little help. Thanks.
The polynomial can be factored over $\Bbb F_7$ as follows $$ x^6+3x^5+2x^4+6x^3+4x^2+5x+2=(x^2 + x + 6)(x^2 + x + 4)(x^2 + x + 3). $$ The shortest way probably is to find first the monic quadratic polynomials which are irreducible, and then use polynomial division.
Attempting to depress your polynomial $f$, it turns out that $$f(x+3)=x^6+1,$$ which is easy to factor by hand.