Example FldNum_norm-equation (H36E14)

We try to solve N(x) = 3 in some relative extension: (Note that since the larger field is a quadratic extension, the second call tells us that there is no integral element with norm 3)

> x := PolynomialRing(Integers()).1;
> K := NumberField([x^2-229, x^2-2]);
> NormEquation(K, 3);
true [
  1/3*K.1 - 16/3
]

Next we solve the same equation but come from a different angle, we will define the norm map as an element of the group ring and, instead of explicitly computing a relative extension, work instead with the implicit fixed field.

> F := AbsoluteField(K);
> t := F!K.2;
> t^2;
2
> A, _, mA := AutomorphismGroup(F);
> S := sub<A | [ x : x in A | mA(x)(t) eq t]>;
> N := map<F -> F | x:-> &* [ mA(y)(x) : y in S]>;
> NormEquation(3, N);
true [
  -5/1*$.1 + 2/3*$.3
]

Finally, to show the effect of Raw:

> f, s, base := NormEquation(3, N:Raw);
> s;
[
  ( 0  1 -1  1 -1  0  2 -1 -1 -1 -1  2  0  0)
]
> z := PowerProduct(base, s[1]);
> z;
-5/1*$.1 + 2/3*$.3
> N(z);
3