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

asked 2020-06-30 20:14:59 -0500

davidac897 gravatar image

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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answered 2020-07-01 20:07:15 -0500

nbruin gravatar image

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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Asked: 2020-06-30 20:14:59 -0500

Seen: 46 times

Last updated: Jul 01