An irreducible polynomial of degree $4$ in $\mathbb{Z}_5[x]$

Question

Q: Find an irreducible polynomial of degree four in $\mathbb{Z}_5[x]$.

My Answer: $x^{4} - 2$

Is my answer correct?

Answer 1

This is correct, but you need to show it's irreducible. Here's one way to do it:

We immediately verify that $x^4-2$ has no roots in $\mathbb Z _5$. Thus, if it is reducible, it must split into two monic quadratic terms. Say $x^4-2=(x^2+ax+b)(x^2+cx+d)=x^4+x^3(a+c)+x^2(b+d+ca)+x(ad+bc)+bd$. Now, matching coefficients, we must have that $bd=-2=3, a+c=0,b+d+ca=0, ad+bc=0$. First, consider $bd=3$. This then holds true for $b=3,d=1$ or $b=4,d=2$. Thus, we must have $b+d+ac=0$ implying that $ac=1$, in the first case, or $ac=4$, in the second. The values for the pair $(a,c)$ are then $(1,4), (2,3), (0,0), (3,2), (4,1)$. Note that there are two pairs each where $ac=1$ or $4$. First, let $(b,d)=(3,1)$, so $(a,c)=(2,3)$ or $(3,2)$. Then $ad+bc=2+9=1$ or $ad+bc= 3+6=9$, neither of which are equal to $0$, so neither of these work. Now, let $(b,d)=(4,2)$, so $(a,c)=(1,4)$ or $(4,1)$. Again, we see $ad+bc = 4+8 =2$ or $16+2=3$, so neither of these work either. Thus, no two quadratic polynomials multiply to $x^4-2$, so it is irreducible.

Answer 2

Adding a shorter solution but one that depends on a basic property of finite fields. The bit I'm gonna need is that their multiplicative groups are cyclic.

You already excluded the possibility of the polynomial $p(x)=x^4-2$ having a linear factor by checking that it has no zeros in the prime field.

Assume that $p(x)$ had an irreducible quadratic factor $q(x)\in\mathbb{Z}_5[x]$. That quadratic factor has a zero $\alpha$ in the field $\mathbb{F}_{25}=\mathbb{Z}_5[x]/\langle q(x)\rangle.$ As $q(\alpha)=0$ then also $p(\alpha)=0$. The element $2$ is of multiplicative order four. Therefore the multiplicative order of $\alpha$ is $16$. But the multiplicative group $\mathbb{F}_{25}^*$ is cyclic of order $24$ and hence cannot have elements of order $16$. This contradicts the existence of an irreducible quadratic factor of $p(x)$.