Irreducible polynomials over $GF(4)$.

Question

I'm working with $GF(4)$ and I'm looking for irreducible polynomials of different degrees over that field. So $GF(4) = \{0,1,\alpha, \alpha + 1\}$ where $\alpha^{2} + \alpha + 1 = 0$, and first I'm looking for an irreducible polynomial of degree 2.

I looked at $\alpha x^2 + x + 1$, and it seems to be irreducible, but in the suggested solution to the problem they have used $x^2 + \alpha x + 1$, which is clearly irreducible. They suggests monic polynomials for the rest of the problems as well, and I'm wondering if they need to be monic?

Answer 1

They don't need to be monic, but you can alway assume they are, up to division by the leading coefficient. Sometimes it's easier to deal with monic polynomials. For example, if you're looking for a polynomial generating some fixed ideal in $\mathbb{F}[X]$, there can be several possible choice, but only one which is monic.

Answer 2

For question of irreducibility (primality) in a domain (and in particular in a UFD), one usually works up to multiplication be an invertible factor. In some cases rather than working with association classes of elements, it is possible and common to impose and additional condition that will select a unique representative from each class; notably in $\Bbb Z$ it is common to only consider positive numbers, and in $K[X]$ it is common to consider only monic polynomials. This is why only monic irreducible polynomials are considered. Their scalar multiples are irreducible as well, but not really of independent interest, just like the irreducible integer $-17$ is not considered independently of the prime$~17$.

In practice this simplifies the classification of the irreducibles considerably. In the current case we can eliminate from the $16$ monic polynomials of degree$~2$, the $X^2+bX+c$ for $b,c\in GF(4)$, the $10$ distinct products of two monic polynomials $X+a$ of degree$~1$ for $a\in GF(4)$. The products involving the irreducible $X$ are the cases with $c=0$, the squares $(X+a)^2=X^2+a^2$ are the cases with $b=0$, and the product $(X+\alpha)(X+\alpha+1)$ is the polynomial $X^2+X+1$ that was irreducible over $GF(2)$ with roots $\alpha,\alpha+1$ in$~GF(4)$. The two remaining reducible polynomials are $(X+1)(X+\alpha)=X^2+(\alpha+1)X+\alpha$ and $(X+1)(X+\alpha+1)=X^2+\alpha X+\alpha+1$. All remaining monic polynomials of degree$~2$ (there are $6$ of them) are irreducible.