Check irreducibility of the polynomial

Question

Problem: It is known that the minimal polynomial of an element $a \in F_4$ equals $x^2 + x + 1$. Does it follow from this that the polynomial $x^3 + ax^2 + a$ is irreducible in the ring of polynomials $F_4$[x]?

Idea: A polynomial of degree three over a field is reducible if and only if it has a root. Since there are only four elements in the field, we can list it: $a, a + 1, 1, 0$. Substituting all these numbers in $p(x) = x^3 + ax^2 + a$, we are convinced that they do not make this polynomial zero. Thus, $p(0) = a, p(1) = 1, p(a) = a, p(a + 1) = a$. But I am not sure whether this solution is correct.

Answer 1

$a+1$ is a root of the given polynomial $P=x^3+ax^2+a$.

$$ \begin{aligned} P(a+1) &= (a+1)^3 +a(a+1)^2+a \\ &=(a+1)^2(\ a+1+a\ ) +a \\ &=(a+1)^2 +a \\ &=(a^2+1^2) +a \\ &=0\ . \end{aligned} $$

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Computer support, sage:

sage: F.<a> = GF(2^2)
sage: a.minpoly()
x^2 + x + 1
sage: P = x^3 + a*x^2 + a 
sage: P.roots()
[(a + 1, 1)]
sage: P(a+1)
0
sage: P.factor()
(x + a + 1) * (x^2 + x + a + 1)