AlgebraicIntegerQ¶

Status: Stable

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

Description¶

AlgebraicIntegerQ[x]

gives True if x is an algebraic integer (a root of a monic polynomial with integer coefficients) and False otherwise. Rational integers are algebraic integers; non-integer rationals and non-algebraic quantities are not. Requires FLINT. Attribute: Protected.

Examples (10)¶

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

Basic examples (4)¶

In[1]:= AlgebraicIntegerQ[Sqrt[2]]
Out[1]= True

In[2]:= AlgebraicIntegerQ[1/2]
Out[2]= False

In[3]:= AlgebraicIntegerQ[(1 + Sqrt[5])/2]
Out[3]= True

In[4]:= AlgebraicIntegerQ[2^(1/3)/2]
Out[4]= False

Applications (6)¶

A root of the monic x^2 - 2

In[5]:= AlgebraicIntegerQ[Sqrt[2]]
Out[5]= True

A non-integer rational is not an algebraic integer

In[6]:= AlgebraicIntegerQ[1/2]
Out[6]= False

A root of the monic x^2 - x - 1

In[7]:= AlgebraicIntegerQ[GoldenRatio]
Out[7]= True

The same number, spelled as a radical

In[8]:= AlgebraicIntegerQ[(1 + Sqrt[5])/2]
Out[8]= True

Gaussian integers are algebraic integers

In[9]:= AlgebraicIntegerQ[2 + 3 I]
Out[9]= True

Scaling by 1/3 leaves the monogenic ring

In[10]:= AlgebraicIntegerQ[Sqrt[2]/3]
Out[10]= False

Implementation notes¶

Algorithm. builtin_algebraicintegerq is a thin wrapper: it checks arity 1 and hands the argument to flint_qqbar_algebraic_integer_q, mapping the tri-state result to True (1), False (0), or unevaluated (−1, FLINT compiled out). The decision is exact. The engine converts x to a FLINT qqbar_t via to_qqbar — the shared converter that accepts integers, rationals, radicals Power[base, p/q], roots of unity, the imaginary unit, Root[] objects and GoldenRatio, combined by + - * / ^. x is an algebraic integer iff the leading coefficient of its primitive integer minimal polynomial is 1 (monic): fmpz_poly_get_coeff_fmpz(lead, QQBAR_POLY(v), deg) and test fmpz_is_one. A rational p/q has minimal polynomial q x − p, so only true integers (q = 1) qualify. Anything that is not a constant algebraic number — a free symbol, Pi, Log[2] — fails the to_qqbar conversion and returns 0 (False), matching WL.

Data structures. The FLINT qqbar_t (an exact algebraic number: its primitive integer minimal polynomial as an fmpz_poly plus an isolating complex enclosure). No AlgebraicNumber[...] object is built — this is a predicate that only reads the minimal polynomial's leading coefficient.

Complexity / limits. Cost is dominated by the to_qqbar conversion of the argument (field arithmetic over the compositum, bounded by the degree cap QQBAR_DEGREE_CAP = 120). Protected, and deliberately not Listable — AlgebraicIntegerQ[list] asks whether the list itself is an algebraic integer (False), not whether its elements are. Declines (stays unevaluated) only when FLINT is unavailable.

Attributes: Protected.

References¶

See also: Pi

H. Cohen, A Course in Computational Algebraic Number Theory, GTM 138 (Springer, 1993), §4.1 (algebraic integers and minimal polynomials).
The FLINT library (https://flintlib.org), qqbar module — exact real and complex algebraic numbers via minimal polynomial plus isolating enclosure.
Source: src/poly/algebraicintegerq.c
Specification: docs/spec/builtins/algebra.md
Tests: tests/test_algebraicnumberdenominator.c
Tests: tests/test_numberfieldintegralbasis.c

Notes & additional examples¶

Notes¶

The test is exact: x is an algebraic integer iff the leading coefficient of its primitive integer minimal polynomial is 1. A rational p/q has minimal polynomial q x - p, so only ordinary integers qualify among the rationals.

Anything that is not a constant algebraic number — a free symbol, Pi, Log[2] — is simply not an algebraic integer, and returns False. The predicate is not Listable: AlgebraicIntegerQ[list] asks about the list itself. Requires FLINT; with FLINT compiled out the call stays unevaluated.