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

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1

May I suggest leaving the original example, and adding the new example in the "Edit" part?

( 2017-12-31 16:54:23 +0200 )edit
( 2018-06-28 07:32:53 +0200 )edit

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There is a simple approach that consists in using number_field_elements_from_algebraics

sage: from sage.rings.qqbar import number_field_elements_from_algebraics
sage: number_field_elements_from_algebraics([alpha, beta])
sage: K, (a,b), phi = number_field_elements_from_algebraics([alpha, beta])
sage: K   # the field
Number Field in a with defining polynomial y^6 + 12*y^4 + 36*y^2 + 707
sage: a   # alpha in K
1/90*a^4 + 1/9*a^2 + 1/2*a + 8/45
sage: b   # beta in K
1/90*a^4 + 1/9*a^2 - 1/2*a + 8/45
sage: phi(a) == alpha  and phi(b) == beta # phi is the embedding K -> QQbar
True

more

This works for this particular case:

sage: P.<x> = QQ[]
sage: f = x^3+2*x+5 # f = P([5,2,0,1]) if you want
sage: f_roots = f.roots(QQbar, multiplicities=False)
sage: f_roots
[-1.328268855668609?,
0.664134427834305? - 1.822971095411114?*I,
0.664134427834305? + 1.822971095411114?*I]
sage: alpha = f_roots[0]
sage: K = QQ[alpha]
sage: K['x'](f).is_irreducible()
False
sage: factors = K['x'](f).factor()
sage: factors
(x - a) * (x^2 + a*x + a^2 + 2)
sage: g = factors[1][0]
sage: g
x^2 + a*x + a^2 + 2
sage: L.<b> = K.extension(g)
sage: L
Number Field in b with defining polynomial x^2 + a*x + a^2 + 2 over its base field
sage: L['x'](g).factor()
(x - b) * (x + b + a)
sage: L['x'](f).factor()
(x - b) * (x - a) * (x + b + a)

more