I'm asked to find an irreducible polynomial of a certain degree in the field Z2.
I won't specify the degree I'm asked here because I'd like to understand the method and know how to apply it to later questions. I only need one irreducible polynomial, I don't need to find them all etc.
Assume that the degree is much too high to look for 2^n different polynomials, or to use polynomial long division to test against irreducible polynomials of degree < n+1/2
Your help and simple explanations would be greatly appreciated!
Edit: Okay thanks for your responses, they're very useful! It seems to get a direct answer I'll need to give the specific degree (however, if anyone could answer to solve with degree n that would be fantastic).
I need to find an irreducible polynomial of degree 20 in Z2. I have an idea but it seems very long-winded:
To find all of the irreducible polynomials of degree 5 then I need to find all the polynomials of degree 5 with no polynomial divisors of degree < 3.
Could I then repeat this to find the irreducible polynomials of degree 10 using the polynomials of degree 5 I found in the previous step, and then repeat again to find the irreducible polynomials of degree 20?
Like I said this seems very long winded, I'm sure there's an easier way.