Trace and Norm

Let \(A\) be an étale algebra over \({{\Bbb Q}}\), with components \(K_1\times\cdots\times K_n\). We define the (absolute) trace on \(A\) as the additive map \({\rm Tr_{A/{{\Bbb Q}}}}\colon A\to {{\Bbb Q}}\) that sends an element \(a\in A\) to \(\sum_{i=1}^n {\rm Tr}_{K_i/{{\Bbb Q}}}(a)\). Let \(m_a\) be the matrix representing the multiplication-by-\(a\) on \(A\) with respect to any basis of \(A\) over \({{\Bbb Q}}\). Then \({\rm Tr}_{A/{{\Bbb Q}}}(a)\) equals the trace of \(m_a\).

We define the (absolute) norm on \(A\) as the multiplicative map \({\rm N}_{A/{{\Bbb Q}}}\colon A \to {{\Bbb Q}}\) by sending a unit \(a \in A\) to \(\prod_{i=1}^n {\rm N}_{K_i/{{\Bbb Q}}}(a)\) and every zero-divisor to \(0\). We have \(N_{A/{{\Bbb Q}}}(a)\) equals the determinant of the matrix \(m_a\).

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 \({\operatorname{Tr}}(x \cdot I)\) is integer-valued.

Let \(I\) be an order or a fractional ideal in an étale algebra \(A\) over \({{\Bbb Q}}\). We defined the trace dual ideal of \(I\) as \(I^t=\{ a\in A : {\rm Tr}_{A/{{\Bbb Q}}}(a\cdot I) \subseteq {{\Bbb Z}}\}\). For fractional ideals \(I\) and \(J\) and a unit \(a\in A\), we have:

(a)

if \(I \subseteq J\) then \(J^t \subseteq I^t\) and \(\#(J/I) = \#(I^t/J^t)\);

(b)

\((aI)^t = {1\over a}I^t\);

(c)

\((I+J)^t = I^t \cap J^t\);

(d)

\((I\cap J)^t = I^t + J^t\);

(e)

\((I:J)^t = I^t\cdot J\).

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 \({\operatorname{Tr}}(x \cdot O)\) is integer-valued.