Rodrigo Raya's profile - activity

Simplify rational expression to polynomial I'd like to check my computations for rational polynomial equations with sage

I'm trying to construct given an irreducible polynomial $f \in \mathbb{Q}[X]$ an extension field that adjoins two of its roots $\alpha_1,\alpha_2$. I'm trying to follow the approach suggested in this question . However, with the following code:

P.<x> = QQ[]
f = x^3+2*x+5 # f = P([5,2,0,0,1]) if you want
f_roots = f.roots(QQbar, multiplicities=False)
print f_roots
alpha = f_roots[0]
beta = f_roots[1]
K = QQ[alpha,beta]
K['x'](f).is_irreducible()

But this gives the error:

ValueError: defining polynomial (x^3 + 2*x + 5) must be irreducible

Although, the polynomial is clearly irreducible over $\mathbb{Q}$. Doing it as:

P.<x> = QQ[]
f = x^3+2*x+5
f_roots = f.roots(QQbar, multiplicities=False)
alpha = f_roots[0]
K.<a> = QQ[alpha]
beta = f_roots[1]
K1.<b> = K[beta]

Gives error:

ValueError: base field and extension cannot have the same name 'a'

What is going wrong? Is this the right way to construct the extension field with two roots?

Edit

Let me emphasize that I'm looking for a method that works for an arbitrary degree not just for degree 3. My actual problem goes on a degree four polynomial. So take as an example:

f = x^4+2*x+5

instead of the previous one.

@slelievre I didn't have the privilege to post a link that's why. i believe the page given in the answer has more information in it:

The output is a 4-component vector [n,s,k,name] with the following meaning: n is the cardinality of the group, s is its signature (s = 1 if the group is a subgroup of the alternating group Ad, s = -1 otherwise) and name is a character string containing name of the transitive group according to the GAP 4 transitive groups library by Alexander Hulpke.

That's the relevant part i think

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.

Take for instance the example:

G = NumberField(x^5 + 15*x + 12, 't').galois_group(type="pari"); G

This gives an output:

Galois group PARI group [20, -1, 3, "F(5) = 5:4"] of degree 5 of the Number Field in t with defining polynomial x^5 + 15*x + 12

So I wonder what is the meaning of [20, -1, 3, "F(5) = 5:4"]. Is it some kind of permutation?

The relevant page of the documentation doesn't clarify it.