Field Theory & Polynomial Structures

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Abstractions about fields and polynomials in algebra and number theory, covering field-extension properties (algebraically closed, formally real, splitting and rupture fields), polynomial classes and operations (cyclotomic, symmetric, and Schubert polynomials, factorization), and arithmetic invariants like fundamental units.

25 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Algebraic number field — A finite-degree field extension of the rational numbers, carrying arithmetic through its ring of integers, embeddings, ideals, norms, traces, units, and completions.
• Algebraically closed field — A field in which every nonconstant univariate polynomial with coefficients in that field has a root, equivalently splits completely into linear factors.
• All one polynomial — A polynomial whose coefficients from degree zero through its leading degree are all one, equivalently (xᵐ⁺¹−1)/(x−1).
• Bernstein's Theorem (Polynomials) — Bernstein's polynomial theorem sharply bounds a complex polynomial's unit-circle derivative norm by its degree times its value norm.
• Binomial (polynomial) — A polynomial consisting of exactly two nonzero monomial terms, whose sparse form supports binomial expansions, toric ideals and algebraic varieties governed by exponent differences.
• Cyclotomic polynomial — The monic irreducible integer polynomial whose roots are exactly the primitive nth roots of unity.
• Degree of a field extension — The vector-space dimension of an extension field over its base field.
• Elimination theory — Remove selected variables from polynomial systems while preserving the algebraic consequences in the retained variables, linking elimination ideals, resultants, Gröbner bases, projection, and implicitization.
• Elliptic unit — A distinguished algebraic unit in an abelian extension of an imaginary quadratic field, constructed from special values of modular or elliptic functions and forming an Euler system.
• Euclidean ordered field — An ordered field in which every nonnegative element has a square root within the field.
• Factorization of polynomials — The decomposition of a polynomial over a specified coefficient domain into a unit and irreducible polynomial factors, unique only under appropriate factorization properties and normalization.
• Formally real field — A field that admits an ordering compatible with its operations, equivalently one in which minus one cannot be expressed as a finite sum of squares.
• Fundamental unit (number theory) — A generator, modulo roots of unity, of the rank-one unit group of a number field's ring of integers.
• Grassmann number — An element of an exterior algebra generated by anticommuting variables, with odd generators squaring to zero.
• Hilbert–Poincaré Series — Encode the dimension or rank of each graded component as the corresponding coefficient of a formal power series.
• Mahler measure — A multiplicative height-like measure of a polynomial equal to its leading coefficient magnitude times the moduli of roots outside the unit circle.
• Minimal polynomial (field theory) — The unique monic polynomial of least positive degree over a base field having a given algebraic extension element as a root.
• Polynomial identity testing — The computational problem of deciding whether an algebraic expression or arithmetic circuit represents the identically zero polynomial rather than merely vanishing at selected inputs.
• Pseudo algebraically closed field — A field in which every absolutely irreducible variety defined over the field has a rational point, imitating a key geometric consequence of algebraic closure without requiring every polynomial to split.
• Quasi-algebraically closed field — Classify a field as C1 when every nonconstant homogeneous form of degree d in more than d variables has a nontrivial zero over that field.
• Rupture field — A field extension generated by adjoining one root of a polynomial over the base field.
• Schubert polynomial — A polynomial indexed by a permutation that represents the corresponding Schubert variety's cohomology class in a flag variety and forms a basis of the polynomial ring in stable variables.
• Separable polynomial — A polynomial over a field whose roots in an algebraic closure are all distinct.
• Splitting field — Form the field extension generated by all roots of a polynomial so that it factors completely into linear terms, with minimality and uniqueness understood relative to the base field.
• Symmetric polynomial — A multivariable polynomial unchanged by every permutation of its variables.