Trace and Norm

Let A be an étale algebra over Q, with components K1 x ... x Kn. We define the (absolute) trace on A as the additive map mathrm(TrA/Q): A to Q that sends an element a∈A to ∑i=1n mathrm(Tr)Ki/Q(a). Let ma be the matrix representing the multiplication-by-a on A with respect to any basis of A over Q. Then mathrm(Tr)A/Q(a) equals the trace of ma.

We define the (absolute) norm on A as the multiplicative map mathrm(N)A/Q: A to Q by sending a unit a ∈A to ∏i=1n mathrm(N)Ki/Q(a) and every zero-divisor to 0. We have NA/Q(a) equals the determinant of the matrix ma.

Trace(x) : AlgEtQElt -> FldRatElt
Returns the trace of the element x of an étale algebra.
Norm(x) : AlgEtQElt -> FldRatElt
Returns the norm of the element x of an étale algebra.
AbsoluteTrace(x) : AlgEtQElt -> FldRatElt
Returns the absolute trace of the element x of an étale algebra. Since the étale algebra is over the rationals this is the same as Trace.
AbsoluteNorm(x) : AlgEtQElt -> FldRatElt
Returns the absolute norm of the element x of an étale algebra. Since the étale algebra is over the rationals this is the same as Norm.
TraceDualIdeal(I) : AlgEtQIdl -> AlgEtQIdl
Returns the trace dual ideal of the ideal I, that is, the set of elements x of the algebra such that Tr(x .I) is integer-valued.

Let I be an order or a fractional ideal in an étale algebra A over Q. We defined the trace dual ideal of I as It={ a∈A : mathrm(Tr)A/Q(a.I) ⊆Z }. For fractional ideals I and J and a unit a∈A, we have:

(a)
if I ⊆J then Jt ⊆It and #(J/I) = #(It/Jt);
(b)
(aI)t = frac(1)(a)It;
(c)
(I + J)t = It ∩Jt;
(d)
(I∩J)t = It + Jt;
(e)
(I:J)t = It.J.
TraceDualIdeal(O) : AlgEtQOrd -> AlgEtQIdl
Returns the trace dual ideal of an order in an étale algebra, that is, the set of elements x of the algebra such that Tr(x .O) is integer-valued.
V2.29, 28 November 2025