Given the polynomial$\ x^4+x+1$, I have to find out if it is irreducible over $\mathbb Q $.
When looking at the solutions, they applied the degree 2 or 3 irreducibly tests to determine that it was irreducible over $\mathbb Q$.
Similarly, they also applied it to the polynomial $x^5+5x^2+1$.
Can someone explain why this is allowed?
The polynomial $x^4+x+1$ is irreducible in $\mathbb{Q}[x]$. That can be proved using this algorithm: one first considers the resolvent cubic of $x^4+x+1$, which is is $x^3-4x-1$. Does it have rational roots? No. Therefore, $x^4+x+1$ is irreducible.
However, the method used for determining whether a quadritic or cubic polynomial is irreducible (determining whether it has a rational root) doesn't work for higher degrees.