Specifically, if m < n, then pi = 0 for m < i ≤ n. The scalar multiplication is the special case of the multiplication where p = p0 is reduced to its constant term (the term that is independent of X); that is It is straightforward to verify that these three operations satisfy the axioms of a commutative algebra over K. Therefore, polynomial rings are also called polynomial algebras.
A straightforward use of the operation rules shows that the expression is then an alternate notation for the sequence Let be a nonzero polynomial with
Most of the properties of K[X] that are listed in this section do not remain true if K is not a field, or if one considers polynomials in several indeterminates.
In particular, two polynomials that are not both zero have a unique greatest common divisor that is monic (leading coefficient equal to 1).
In the case of the integers the same property is true, if degrees are replaced by absolute values, but, for having uniqueness, one must require a > 0.
If K is the field of complex numbers, the fundamental theorem of algebra asserts that a univariate polynomial is irreducible if and only if its degree is one.
Already for the integers, there is no known algorithm running on a classical (non-quantum) computer for factorizing them in polynomial time.
In the case of the real or complex numbers, Abel–Ruffini theorem shows that the roots of some polynomials, and thus the irreducible factors, cannot be computed exactly.
If the evaluation homomorphism is not injective, this means that its kernel is a nonzero ideal, consisting of all polynomials that become zero when X is substituted with θ.
In linear algebra, the n×n square matrices over K form an associative K-algebra of finite dimension (as a vector space).
In fact, if p is irreducible, every nonzero polynomial q of lower degree is coprime with p, and Bézout's identity allows computing r and s such that sp + qr = 1; so, r is the multiplicative inverse of q modulo p. Conversely, if p is reducible, then there exist polynomials a, b of degrees lower than deg(p) such that ab = p ; so a, b are nonzero zero divisors modulo p, and cannot be invertible.
For example, the standard definition of the field of the complex numbers can be summarized by saying that it is the quotient ring and that the image of X in
The structure theorem for finitely generated modules over a principal ideal domain applies to K[X], when K is a field.
This is commonly used for proving properties of multivariate polynomial rings, by induction on the number of indeterminates.
Polynomial rings in several variables over a field are fundamental in invariant theory and algebraic geometry.
In particular, because of the geometric applications, many interesting properties must be invariant under affine or projective transformations of the indeterminates.
Bézout's theorem, Hilbert's Nullstellensatz and Jacobian conjecture are among the most famous properties that are specific to multivariate polynomials over a field.
and the geometric properties of algebraic varieties, that are (roughly speaking) set of points defined by implicit polynomial equations.
The first version generalizes the fact that a nonzero univariate polynomial has a complex zero if and only if it is not a constant.
has a common zero in an algebraically closed field containing K, if and only if 1 does not belong to the ideal generated by S, that is, if 1 is not a linear combination of elements of S with polynomial coefficients.
One can also consider a strictly larger ring, by defining as a generalized polynomial an infinite (or finite) formal sum of monomials with a bounded degree.
The formulas for addition and multiplication make sense as long as one can add exponents: Xi ⋅ Xj = Xi+j.
A set for which addition makes sense (is closed and associative) is called a monoid.
The set of functions from a monoid N to a ring R which are nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of N with coefficients in R. The addition is defined component-wise, so that if c = a + b, then cn = an + bn for every n in N. The multiplication is defined as the Cauchy product, so that if c = a ⋅ b, then for each n in N, cn is the sum of all aibj where i, j range over all pairs of elements of N which sum to n. When N is commutative, it is convenient to denote the function a in R[N] as the formal sum: and then the formulas for addition and multiplication are the familiar: and where the latter sum is taken over all i, j in N that sum to n. Some authors such as (Lang 2002, II,§3) go so far as to take this monoid definition as the starting point, and regular single variable polynomials are the special case where N is the monoid of non-negative integers.
Polynomials in several variables simply take N to be the direct product of several copies of the monoid of non-negative integers.
Several interesting examples of rings and groups are formed by taking N to be the additive monoid of non-negative rational numbers, (Osborne 2000, §4.4).
Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms.
A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained.
Extending this relation by associativity and distributivity allows explicitly constructing the Weyl algebra.
(Lam 2001, §1,ex 1.11) Skew polynomial rings are closely related to crossed product algebras.