Ask Your Question

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

asked 2017-12-30 19:14:54 +0200

Rodrigo Raya gravatar image

updated 2017-12-30 20:04:30 +0200

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?


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.

edit retag flag offensive close merge delete


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

slelievre gravatar imageslelievre ( 2017-12-31 00:16:49 +0200 )edit

1 Answer

Sort by ยป oldest newest most voted

answered 2017-12-31 00:25:25 +0200

slelievre gravatar image

You could do something like the following.

$ sage -v
SageMath version 8.1, Release Date: 2017-12-07
$ sage -q
sage: P.<x> = QQ[]
sage: f = P([QQ.random_element() for _ in range(6)])
sage: f
x^4 - 2*x^3 - x^2 + x + 8
sage: f.roots()
sage: f_roots = f.roots(QQbar, multiplicities=False)
sage: f_roots
[-0.9308781364905225? - 0.9859665944129413?*I,
 -0.9308781364905225? + 0.9859665944129413?*I,
 1.930878136490523? - 0.7891098514623471?*I,
 1.930878136490523? + 0.7891098514623471?*I]
sage: alpha = f_roots[0]
sage: K = QQ[alpha]
sage: K
Number Field in a with defining polynomial x^4 - 2*x^3 - x^2 + x + 8
sage: K['x'](f).factor()
(x - a) * (x^3 + (a - 2)*x^2 + (a^2 - 2*a - 1)*x + a^3 - 2*a^2 - a + 1)
edit flag offensive delete link more

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools



Asked: 2017-12-30 19:14:54 +0200

Seen: 611 times

Last updated: Dec 31 '17