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Absolute number field extension

asked 2022-02-21 00:27:42 +0200

anonymous user


updated 2022-02-21 22:45:54 +0200

slelievre gravatar image

Hi there,

I am working with the following absolute field extension of a quadratic field.

d =  # some integer
K.(a) = NumberField(x^2-d)
R.<x> = PolynomialRing(K)
g = x^2 + x + 1
KL.(b) = K.extension(g)
f = KL.absolute_polynomial()
L.(c) = NumberField(f)

I need to keep track of the element b inside L and I don't know how. Any help is much appreciated.

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answered 2022-02-22 13:07:43 +0200

rburing gravatar image

Use the absolute_field method instead:

sage: L.<c> = KL.absolute_field()
sage: f = L.defining_polynomial()
sage: from_L, to_L = L.structure()
sage: to_L(b)
2/11*c^3 + 3/11*c^2 + 2/11*c - 5/11

(The output is for the choice of d = 2.)

You also get the morphism in the other direction, which can be handy:

sage: from_L(c)
b - a

(Again, the output is for the choice of d = 2.)

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