MinimalPolynomial

Status: Stable

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

Description

MinimalPolynomial[s, x]
    gives the lowest-degree polynomial in x with integer coefficients,
    positive leading coefficient and content 1, having the algebraic
    number s as a root.  s may be built from rationals, radicals, the
    imaginary unit, roots of unity, and Root[] objects.
MinimalPolynomial[s]
    gives the minimal polynomial as a pure function.
MinimalPolynomial[s, x, Extension -> a]
    gives the characteristic polynomial of s in Q(a) over Q(a).
    Computed by resultant elimination of the radicals; threads over
    lists.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= MinimalPolynomial[Sqrt[2] + Sqrt[3], x]
Out[1]= 1 - 10 x^2 + x^4

In[2]:= MinimalPolynomial[(1 + I)/Sqrt[2], x]
Out[2]= 1 + x^4

In[3]:= MinimalPolynomial[Root[2 #1^3 - 2 #1 + 7 &, 1] + 17, x]
Out[3]= -9785 + 1732 x - 102 x^2 + 2 x^3

In[4]:= MinimalPolynomial[Sqrt[2], x, Extension -> E^(I Pi/4)]
Out[4]= 4 - 4 x^2 + x^4

Implementation notes

Algorithm. builtin_minimalpolynomial computes the minimal polynomial of an algebraic number s over Q in x via resultant elimination:

1. Atom walk (mp_walk): recursively rewrite s into a "value expression" V in fresh auxiliary symbols t_i, recording for each t_i a polynomial defining relation p_i (in t_i and earlier auxiliaries). Every relation is kept polynomial — negative powers are turned into reciprocal variables (D·w - 1) so no fractions appear.
2. Elimination: build g = (x - V) and eliminate each t_i (highest index first) by g <- Resultant[g, p_i, t_i]. The introduction order guarantees t_i is present in g when eliminated and that each relation references only earlier auxiliaries, so the chain terminates in a univariate G(x).
3. Clear and factor: take Numerator[Together[G]], make primitive over Z, and factor. Evaluate s numerically to high precision and select the unique irreducible factor that vanishes at s.
4. Return that factor, primitive with positive leading coefficient.

Extension -> a uses the tower law: if s ∈ Q(a), the characteristic polynomial of s over Q(a) is m_s(x)^([Q(a):Q]/[Q(s):Q]); membership is checked via the primitive-element degree [Q(a,s):Q] == [Q(a):Q].

Data structures. Expr* trees with fresh internal auxiliary symbols; resultants run on the multivariate polynomial machinery (Resultant), denominator clearing through internal_together/Numerator, primitivisation/factoring via the Z-polynomial routines (zupoly, facpoly), and root selection through high-precision numericalize (MPFR when built). The builtin takes ownership of res and returns a fresh Expr* or NULL.

Complexity / limits. Dominated by the iterated resultant elimination (one resultant per auxiliary symbol, with the usual degree blow-up) and the final univariate factorisation. Numeric root-matching disambiguates the irreducible factor.

• Listable, Protected. A List first argument threads element-wise.
• s may be built from integers and rationals, radicals (Sqrt,

Attributes: Listable, Protected.

Implementation status

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

References

• Source: src/poly/minpoly.c
• Specification: docs/spec/builtins/algebra.md

Notes & additional examples

Worked examples

In[1]:= MinimalPolynomial[Sqrt[2], x]
Out[1]= -2 + x^2

In[1]:= MinimalPolynomial[(1 + Sqrt[5])/2, x]
Out[1]= -1 - x + x^2

In[1]:= MinimalPolynomial[Sqrt[2] + Sqrt[3], x]
Out[1]= 1 - 10 x^2 + x^4

In[1]:= MinimalPolynomial[Cos[2 Pi/5], x]
Out[1]= -1 + 2 x + 4 x^2

In[1]:= MinimalPolynomial[Sqrt[2 + Sqrt[2]], x]
Out[1]= 2 - 4 x^2 + x^4

Notes

MinimalPolynomial[s, x] returns the lowest-degree integer polynomial in x, with positive leading coefficient and content 1, that has the algebraic number s as a root. The first two examples recover the defining polynomials of Sqrt[2] and the golden ratio (x^2 - x - 1). The real power shows up with compound algebraic numbers: Sqrt[2] + Sqrt[3] is degree 4 over the rationals, and MinimalPolynomial finds its quartic x^4 - 10 x^2 + 1 by eliminating the radicals with resultants — not by numerical root-finding. It also handles algebraic constants beyond plain radicals: Cos[2 Pi/5] (a root of a cyclotomic relation) yields 4 x^2 + 2 x - 1, and the nested radical Sqrt[2 + Sqrt[2]] yields the quartic x^4 - 4 x^2 + 2. The input may be built from rationals, radicals, the imaginary unit, roots of unity, and Root[] objects.