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Define morphism of Q[i,w] in terms of i and w

asked 2021-09-06 10:35:27 +0200

JNS gravatar image

I want to construct the field $K=Q[i,w]$ where $w=\sqrt[4]{2}$ and define a field homomorphism such as $w\mapsto iw$, $i\mapsto -i$. I'd like to do

K.<w,I> = NumberField([x^4-2,x^2+1])
H = End(K)

but it doesn't work because $K$ is considered to be the relative field $(Q[i])[w]$. I know I could work with the absolute field

K_abs.<theta> = K.absolute_field()

but I'd really like to define the morphism in terms of $w,i$. Is it possible?

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answered 2021-09-06 14:36:38 +0200

rburing gravatar image

When you construct the tower of extensions explicitly, you can do the following:

sage: K.<I> = NumberField(x^2+1)
sage: L.<w> = K.extension(x^4-2)
sage: f = L.hom([I*w], base_map=K.hom([-I])); f
Relative number field endomorphism of Number Field in w with defining polynomial x^4 - 2 over its base field
  Defn: w |--> I*w
        I |--> -I
sage: f(I + w)
I*w - I
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Asked: 2021-09-06 10:35:27 +0200

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Last updated: Sep 06 '21