# Number Fields and the Norm

Hello,

I am currently very confused about how to use [Number Fields in Sage].(http://www.sagemath.org/doc/reference/sage/rings/number_field/number_field.html). Things are not working.

Specifically I would like the norm of $\mathbb{Z}[\zeta_5]$ which should be a homogeneous polynomial of degree 4.

I tried:

K.<g> = NumberField(1+x+x^2+x^3+x^4)

a,b,c,d=var('a b c d')

(a+b*g+c*g^2+d*g^3).norm()

However that did not work.

I also tried:

K.<g>=CyclotomicField(5); K

a,b,c,d=var('a b c d')

(a+b*g+c*g^2+d*g^3).norm()

Instead of returning the norm in the Cyclotomic ring, this simply gave me the norm over the complex numbers, that is multiplication by the complex conjugate. These two things are very different, and the norm of the complex number is not at all correct. (It should be degree 4 not degree 2)

Any help is greatly appreciated, thank you