Polynomial RingEdit
Polynomial rings are a foundational construction in abstract algebra, built from a given ring and a single indeterminate. They provide a bridge between the algebra of coefficients and the algebra of polynomials in one symbol, encapsulating both addition and a natural notion of multiplication through the distributive law. When the coefficient ring is commutative, the polynomial ring inherits commutativity and becomes a central object of study in commutative algebra and beyond. For more general settings, one can also form polynomial rings in several indeterminates, leading to the multivariate theory that underpins much of algebraic geometry and algebraic combinatorics.
The story of polynomial rings begins with the idea of taking linear expressions in a new symbol x with coefficients from a given ring R. The resulting expressions form a new ring, denoted R[x], whose elements are finite sums a0 + a1 x + a2 x^2 + … + an x^n with coefficients ai in R. Addition is done by adding coefficients termwise, while multiplication uses the distributive rule and the rule x·a = a·x for all a in R. If R is commutative, then x commutes with every coefficient, and R[x] is itself a commutative ring. If R is not commutative, one typically imposes that x be a central indeterminate to obtain a well-behaved polynomial ring in one central variable.
Definition
Let R be a ring with unity. The polynomial ring R[x] consists of all polynomials f = ∑_{i=0}^m a_i x^i with finitely many nonzero coefficients a_i ∈ R. Addition is defined coefficientwise, and multiplication is defined by the rule x^i · x^j = x^{i+j} together with the distributive law. When R is commutative, R[x] is a commutative ring and contains R as the subring of constant polynomials (a ↦ a). The symbol x is called an indeterminate or a formal variable, and it is treated as a symbol that commutes with the coefficients if R is commutative. The construction generalizes to several indeterminates, giving the multivariate polynomial ring R[x1, x2, ..., xn].
In many contexts, one emphasizes a universal property: for any ring homomorphism f: R → S and any element s in the center of S (i.e., commuting with f(R)), there exists a unique ring homomorphism φ: R[x] → S with φ|_R = f and φ(x) = s. This universal property expresses R[x] as the free commutative R-algebra on one generator. See universal property for a broader discussion of this viewpoint.
Basic properties
Structure: R[x] is a ring with unity. The map R → R[x], a ↦ a, embeds R as the constant polynomials.
Commutativity: If R is commutative, then R[x] is commutative. In general, R[x] is noncommutative if R is noncommutative, unless x is constrained to be central.
Degree: One assigns to a nonzero polynomial f = ∑ a_i x^i the degree deg(f) = max{i : a_i ≠ 0}, and deg(0) is defined as −∞ (often treated as a special symbol). This makes R[x] a graded ring.
Ideals: The structure of ideals in R[x] reflects both R and the polynomial variable. If R is a field (i.e., a field), then R[x] is a principal ideal domain (PID) and a Euclidean domain, with the degree function providing the Euclidean algorithm. In particular, when R is a field k, the ideals of k[x] are exactly the principal ideals generated by polynomials, and every nonzero ideal is generated by a monic polynomial. See principal ideal domain and Euclidean domain for related concepts.
Factorization: If R is a field, k[x] is a unique factorization domain (UFD); Gauss’s lemma supplies a bridge between factorization in k[x] and factorization of polynomials with coefficients in k. See unique factorization domain.
Primes and irreducibles: In k[x], irreducible polynomials generate prime ideals, and prime ideals in k[x] are well understood: they are either (0) or (p) where p is irreducible. This picture becomes more intricate over general R.
Evaluation and roots: For a ∈ R, substituting x ↦ a defines an evaluation homomorphism f ↦ f(a) from R[x] to R (when R is appropriate for the context). Roots of polynomials connect to solutions of equations in the base ring, a theme that runs through many areas of algebra and algebraic geometry.
Multivariate case: In R[x1, x2, ..., xn], monomials are of the form x1^{e1} x2^{e2} … xn^{en}, and the degree is the total degree ∑ ei. This setting is central to the study of multivariate algebraic objects and their geometric counterparts, such as affine varieties.
Variants and extensions
Multivariate polynomial rings: As noted, R[x1, x2, ..., xn] generalizes the univariate case to finitely many indeterminates. This extension preserves many properties from the univariate case and interacts richly with algebraic geometry via solutions to polynomial equations in several variables.
Noncommutative variants: If R is noncommutative, one can form polynomial rings with x central to obtain a central polynomial ring, or use more general constructions such as Ore extensions to model noncommuting variables together with automorphisms or derivations of R.
Applications to modules and algebras: Polynomial rings serve as a convenient ambient space for building modules, algebras, and representations. For example, one studies R[x]-modules by looking at how x acts; this viewpoint is fundamental in areas like Noetherian theory and representation theory.
Connections to computational algebra: Algorithms for factoring polynomials, computing gcds, and performing division with remainder in R[x] are central to computer algebra systems and symbolic computation. The behavior of polynomials over fields is especially well understood and forms a testing ground for algorithms in algorithmic algebra.
Applications and significance
Polynomials with coefficients in a given ring encode a wide range of problems, from solving equations to modeling geometric objects. The univariate case over a field provides a clean and well-understood theory of factorization, roots, and linear recurrences, while the multivariate case opens doors to algebraic geometry and the study of shapes defined by polynomial equations. The polynomial ring also interacts with concepts like modules, ideals, and homomorphisms, making it a versatile tool across abstract algebra and its applications.