ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sun, 31 Dec 2017 02:49:21 +0100Extension field adjoining two rootshttps://ask.sagemath.org/question/40389/extension-field-adjoining-two-roots/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](https://ask.sagemath.org/question/40374/factorization-of-f-in-mathbbqx-in-field-extension-mathbbqalpha/)
. 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.Rodrigo RayaSun, 31 Dec 2017 02:49:21 +0100https://ask.sagemath.org/question/40389/