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polynomial

Contemts

Definition

Let R be a commutative ring.

Example

The set of polynomials in one variable with coefficients in R is the set R[] of all formal linear combinations on elements n, thought of as powers x n of the variable x

a nz n++a 1z+a 0,a_n z^n + \cdots + a_1 z + a_0 \,,

where n is an arbitrary natural number and a 0,,a nR, modulo the equivalence relation generated by equations of the form

0z n+1+a nz n++a 1z+a 0=a nz n++a 1z+a 00 z^{n+1} + a_n z^n + \cdots + a_1 z + a_0 = a_n z^n + \cdots + a_1 z + a_0

(so that we ignore coefficients of zero).

This set is equipped with the structure of a ring itself, in fact a commutative algebra over R, denoted R[z] and called the polynomial ring or ring of polynomials given by the unique bilinear map

R[z]R[z]R[z]R[z] \cdot R[z] \to R[z]

which on monomials is given by

z kz l=z k+l.z^k \cdot z^l = z^{k+l} \,.

Properties

Proposition

The polynomial ring R[z] is the free R-algebra on one generator (the variable z).

Proof

By the definition of free objects one needs to check that ring homomorphisms

f:R[z]Kf : R[z] \to K

to another ring K are in natural bijection with functions of sets

f¯:*K\bar f : * \to K

from the singleton to the set underlying K. Take f¯f(z). Using R-linearity, this is directly seen to yield the desired bijection.

Remark

Similarly, the set of polynomials in any give set of variables with coefficients in R is the free commutative R-algebra on that set of generators; see symmetric power and symmetric algebra.

Remark

The field of fractions of R[z] is the field R(z) of rational functions.

Revised on October 2, 2012 12:58:29 by Urs Schreiber (131.174.41.94)