# Factorization of $f \in \mathbb{Q}[X]$ in field extension $\mathbb{Q}(\alpha)$.

I'm given an irreducible polynomial $f \in \mathbb{Q}[X]$ of degree 5 and I want to determine its Galois group without using the predefined functions of sage. The method I want to follow takes a root $\alpha_1$ of $f$ and studies the factorization of $f$ in the field extension $\mathbb{Q}(\alpha_1)$.

I believe this is possible with other software. How can I do it with sage?

**Edit**

Apparently, Abstract Algebra: An Interactive Approach, Second Edition but they use InitDomain function which is not recognized by my notebook.

Apparently the book gives a CD where an interface between sage and gap is done. So probably the solution requires using gap commands.

It helps if you give an explicit example to get others started exploring your question.