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

Processing math: 100%

JNS
25 ●4

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?

add a comment

1 Answer

Sort by » oldest newest most voted

answered 3 years ago

rburing

10964 ●4 ●80 ●219 https://www.rburing.nl/

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

link

add a comment

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

1 follower

Stats

Asked: 3 years ago

Seen: 255 times

Last updated: Sep 06 '21

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

1 Answer