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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?

Define morphism of Q[i,w] in terms of i and w I want to construct the field $K=Q[i,w]$ where $w=\sqrt[4]{2}$ and define

I have a problem with creating a rational map from the projective plane to P^1xP^1. The following code gives the error "polys (=[x, y, x^2, y^2]) must be of the same degree":

K = GF(2)
P2.<x,y,z> = ProjectiveSpace(K,2)
P1P1.<x0,x1,y0,y1> = ProductProjectiveSpaces([1,1],K)
H = Hom(P2,P1P1)
H([x,y,x^2,y^2])

On the other hand, creating the "same" map on P^1xP^1 does not give an error:

E = End(P1P1)
E([x0,x1,x0^2,x1^2])

Can someone explain to me why it does not work and how I can go around this problem?