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The polynomials $x^4+x^3+1$ and $x^4+x^3+x^2+x+1$ are irreducibles over GF(2).

(a) Factor both polynomials into irreducibles over GF(4). (b) Factor both polynomials into irreducibles over GF(8).

I simply don't have a clue how to do that. Can someone please give me a hand?

Thanks in advance.

someGal
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  • Can you at least see why they have no linear factors and no trinomial factors in a)? The only remaining cases that could occur are "already irreducible" or "product of two quadratics." To do the $GF(8)$ version, it might be good to build a table for $GF(8)$. – rschwieb Jan 20 '14 at 21:06
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    You didn’t say what you already know about the Galois theory of finite fields. If I knew nothing, I would take @rschwieb’s suggestion and write out all possible quadratic irreducibles over $\mathbb F_4$, not as difficult as it sounds. Then just sit down and do the divisions. Part (b) is very easy if you know the Galois theory. – Lubin Jan 20 '14 at 21:34
  • Agree with professor Lubin - Galois theory is your friend. For the necessary field tables see this Q&A provided for just you by yours truly:-) Surely you recognize the latter polynomial as the fifth cyclotomic polynomial. The former is the reciprocal for the (possible more common) primitive polynomial used in the log-tables. So its zeros are reciprocals of the zeros of $x^4+x+1$. – Jyrki Lahtonen Jan 20 '14 at 21:45

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