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.Mon, 06 Sep 2021 14:36:38 +0200Define morphism of Q[i,w] in terms of i and whttps://ask.sagemath.org/question/58849/define-morphism-of-qiw-in-terms-of-i-and-w/ 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)
H([I*w,-I])
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?
Mon, 06 Sep 2021 10:35:27 +0200https://ask.sagemath.org/question/58849/define-morphism-of-qiw-in-terms-of-i-and-w/Answer by rburing for <p>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</p>
<pre><code>K.<w,I> = NumberField([x^4-2,x^2+1])
H = End(K)
H([I*w,-I])
</code></pre>
<p>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</p>
<pre><code>K_abs.<theta> = K.absolute_field()
</code></pre>
<p>but I'd really like to define the morphism in terms of $w,i$. Is it possible?</p>
https://ask.sagemath.org/question/58849/define-morphism-of-qiw-in-terms-of-i-and-w/?answer=58856#post-id-58856When 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
Mon, 06 Sep 2021 14:36:38 +0200https://ask.sagemath.org/question/58849/define-morphism-of-qiw-in-terms-of-i-and-w/?answer=58856#post-id-58856