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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?

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Here $K$ is the field with two elements, I reckon? – Andreas Caranti Oct 5 '13 at 19:47
    
Don't need to work so hard. As long as you don't have divisibility by $x+1$ or $x^2+x+1$, you have irreducibility. – André Nicolas Oct 5 '13 at 19:50
    
yes its 0s and 1s. @AndréNicolas how can it be proven? – mjanisz1 Oct 5 '13 at 19:52
1  
Because if you write $5$ as the sum of two smaller positive integers, one of them will be $1$ or $2$. – Andreas Caranti Oct 5 '13 at 19:52

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