AlgebraicNumberTrace¶

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description¶

AlgebraicNumberTrace[a]

gives the field trace of the algebraic number a: the sum of the conjugates of a over the rationals, equivalently the sum of the roots of a's minimal polynomial. a may be an integer, a rational, a radical, GoldenRatio, a Root object, or an AlgebraicNumber object. AlgebraicNumberTrace[a, Extension -> theta] gives the trace relative to the field Q(theta), for a an element of Q(theta). Threads over lists. Requires FLINT. Attributes: Listable, Protected.

Examples (10)¶

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (4)¶

In[1]:= AlgebraicNumberTrace[5 + Sqrt[2]]
Out[1]= 10

In[2]:= AlgebraicNumberTrace[GoldenRatio]
Out[2]= 1

In[3]:= AlgebraicNumberTrace[Root[#1^4 + 11 #1^3 + #1^2 + #1 + 1 &, 1]]
Out[3]= -11

In[4]:= AlgebraicNumberTrace[{5 + Sqrt[2], E^(Pi I/8)}]
Out[4]= {10, 0}

Options (2)¶

In[5]:= AlgebraicNumberTrace[5, Extension -> Sqrt[2]]
Out[5]= 10

In[6]:= AlgebraicNumberTrace[E^(2 Pi I/3), Extension -> ToNumberField[{2^(1/3), E^(2 Pi I/3)}, All][[1, 1]]]
Out[6]= -3

Applications (4)¶

Sqrt[2] + (-Sqrt[2]) = 0

In[7]:= AlgebraicNumberTrace[Sqrt[2]]
Out[7]= 0

(1 + Sqrt[2]) + (1 - Sqrt[2])

In[8]:= AlgebraicNumberTrace[1 + Sqrt[2]]
Out[8]= 2

The trace of the golden ratio

In[9]:= AlgebraicNumberTrace[GoldenRatio]
Out[9]= 1

Trace of 1 + 3 Sqrt[2]

In[10]:= AlgebraicNumberTrace[AlgebraicNumber[Sqrt[2], {1, 3}]]
Out[10]= 2

Algorithm¶

algebraicnumbertrace.c — AlgebraicNumberTrace[a].

See algebraicnumbertrace.h.  The value — the field trace of an algebraic number,

optionally relative to a field Q(theta) given by Extension -> theta — is computed exactly by flint_qqbar_algebraic_number_trace (both cases reduce to a

minimal-polynomial read; see flint_qqbar.c).  This file only separates the

positional argument from the Extension option, checks arity, maps the tri/quad-state engine result to Integer/Rational / message / unevaluated, and (being Listable) lets the evaluator thread over a list of algebraic numbers.

Implementation notes¶

Algorithm. builtin_algebraicnumbertrace mirrors AlgebraicNumberNorm structurally: it separates a trailing Extension -> theta option, checks arity, and delegates to flint_qqbar_algebraic_number_trace. The absolute trace Tr_{Q(a)/Q}(a) is the sum of the roots of a's primitive integer minimal polynomial P(x) = c_n x^n + … + c_0, equal to −c_{n−1}/c_n (qqbar_abs_trace). With Extension -> theta the relative trace Tr_{Q(theta)/Q}(a) follows from transitivity of the trace in the tower Q ⊆ Q(a) ⊆ K = Q(theta):

Tr_{K/Q}(a) = [K:Q(a)] · Tr_{Q(a)/Q}(a) = (n/d) · (absolute trace)

with n = deg minpoly(theta), d = deg minpoly(a), and a ∈ Q(theta) decided by qqbar_express_in_field. The contrast with the norm is exact: the trace is additive, so it scales by the tower index n/d, whereas the multiplicative norm is raised to the power n/d. Return codes map to the AlgebraicNumberTrace::ext (a not in Q(theta)) and AlgebraicNumberTrace::nalg (not a constant algebraic number) messages through mth_message, a success Integer/Rational, or a silent decline (FLINT off).

Data structures. FLINT qqbar_t for a and theta, an fmpq accumulator for the trace, and an fmpq_poly for membership in the relative case.

Complexity / limits. O(1) minimal-polynomial read for the absolute trace; the relative case adds a primitive-element build and an express-in-field solve. Listable, Protected; bounded by the degree cap QQBAR_DEGREE_CAP = 120.

Attributes: Listable, Protected.

References¶

H. Cohen, A Course in Computational Algebraic Number Theory, GTM 138 (Springer, 1993), §4.3 (norm and trace of algebraic numbers; transitivity in a tower).
The FLINT library (https://flintlib.org), qqbar module — minimal polynomial and field membership.
Source: src/poly/algebraicnumbertrace.c
Specification: docs/spec/builtins/algebra.md
Tests: tests/test_algebraicnumbertrace.c

Notes & additional examples¶

Notes¶

The absolute trace Tr_{Q(a)/Q}(a) is the sum of the conjugates of a — equivalently the sum of the roots of a's minimal polynomial, read off as -(coeff of x^{deg-1}) / (leading coeff). AlgebraicNumberTrace[a, Extension -> theta] gives the relative trace Tr_{Q(theta)/Q}(a) for a in Q(theta).

Where the norm is multiplicative and raises the absolute value to the tower index, the trace is additive and scales by it. Listable and Protected; requires FLINT.