Need some explanation and checking if my thinking on the solution is correct for the assignment given below:
(In these problems you may use without proof which polynomials of degree 2 and 3 are irreducible in $K[x]$)
Find all the irreducible polynomials in $K[x]$ that are of the form $ 1 + a_1x + x ^2 + a_3x ^3 + a_4x ^4 + x ^5 $ .
So my method was to divide it by all possible irreducible polynomials of degree $<5$. So at first I divided it by $x+1$ and got the reminder $1+a_4+a_3+a_1$. To make it irreducible the reminder has to be greater than $0$. Now I have to divide by rest of polynomials of degree $<5$ so $x , x^2+x+1,x^3+x^2+1,x^3+x+1$. How to get a polynomial of degree 4 and prove its irreducible? also divide it by these polynomials?
Second task is: Factorize $1 + x + x ^4 + x ^7$ into irreducible factors in $K[x]$. I've started dividing it by $(x+1)$ till $(x+1)^2(x^5+x^3+x^2+x+1)$. Now I need to try dividing it as in previous case by other polynomials of degree $<5$ or is there some other method to do it faster?