# Is there a way to compute the norm form of a number ring?

I have a cubic number field $M$, and I want to find the norm form of its integer ring (as a degree $3$ polynomial in $3$ variables). Does this functionality exist in SAGE?

For now, I found this solution:

O = M.maximal_order()
RRR.<a,b,c> = PolynomialRing(M)
lists = []
d = 1

for i in O.gens():
lists.append(i.galois_conjugates(M))

for i in range(3):
e = 0
e += a*lists[0][i]
e += b*lists[1][i]
e += c*lists[2][i]
d *= e

d

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The easiest way is probably to write the linear factor that the norm form splits off over the number field and compute its norm (which is the norm form) as a resultant taken with the minimal polynomial of the field generator:

sage: M.<a>=NumberField(x^3-x-8)
sage: B=[M(a) for a in M.maximal_order().basis()]
sage: R.<x0,x1,x2,a>=QQ[]
sage: f=sum([R.gen(i)*B[i].lift()(a) for i in [0,1,2]])
sage: f.resultant(a^3-a-8,a)
x0^3 + x0^2*x1 - 6*x0*x1^2 + 8*x1^3 + 2*x0^2*x2 - 11*x0*x1*x2 + 44*x1^2*x2 + x0*x2^2 + 92*x1*x2^2 + 64*x2^3


You should generally try to avoid working over splitting fields. It's rarely necessary and, for larger degree extensions, often infeasible.

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