Degree of all extensions $\mathbb Q(\alpha)/\mathbb Q$ with $\alpha$ root of $T^5-T^4+3T^3+3T^2-9T+3$

Question

I need to find the degree of all extensions $\mathbb Q(\alpha)/\mathbb Q$ with $\alpha$ root of $T^5-T^4+3T^3+3T^2-9T+3$.

This polynomial has only one rational solution ($\alpha=1$) and the others are not easy to find.

Even if the 5 roots were easy to get, is there a way of finding those degrees without calculating them all?

I think I have a brief idea of the answer, because this poly is reducible into two factors (degree 1 and 4). Both are monic and irreducible, so the two extensions should be 1 and 4. Am I right?

Thanks in advance.

Answer 1

The degree of an extension $L/K$ with $L$ generated by a single element $\alpha$ is the degree of the minimal polynomial of $\alpha$. The minimal polynomial must divide $T^5-T^4+3T^3+3T^2-9T+3$ and be irreducible. So, yes, you're correct! As the irreducible factors have degree $1$ and $4$, these are the two possibilities.