Polynomial - Abstract Algebra

Abstract Algebra

In abstract algebra, one distinguishes between polynomials and polynomial functions. A polynomial f in one variable X over a ring R is defined as a formal expression of the form

where n is a natural number, the coefficients are elements of R, and X is a formal symbol, whose powers Xi are just placeholders for the corresponding coefficients ai, so that the given formal expression is just a way to encode the sequence, where there is an n such that ai = 0 for all i > n. Two polynomials sharing the same value of n are considered equal if and only if the sequences of their coefficients are equal; furthermore any polynomial is equal to any polynomial with greater value of n obtained from it by adding terms in front whose coefficient is zero. These polynomials can be added by simply adding corresponding coefficients (the rule for extending by terms with zero coefficients can be used to make sure such coefficients exist). Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term aiXi is interpreted as a polynomial that has zero coefficients at all powers of X other than Xi. Then to define multiplication, it suffices by the distributive law to describe the product of any two such terms, which is given by the rule

 a X^k \; b X^l = ab X^{k+l} for all elements a, b of the ring R and all natural numbers k and l.

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R. The map from R to R sending r to rX0 is an injective homomorphism of rings, by which R is viewed as a subring of R. If R is commutative, then R is an algebra over R.

One can think of the ring R as arising from R by adding one new element X to R, and extending in a minimal way to a ring in which X satisfies no other relations than the obligatory ones, plus commutation with all elements of R (that is Xr = rX). To do this, one must add all powers of X and their linear combinations as well.

Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring R over the real numbers by factoring out the ideal of multiples of the polynomial X2 + 1. Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring R (see modular arithmetic).

If R is commutative, then one can associate to every polynomial P in R, a polynomial function f with domain and range equal to R (more generally one can take domain and range to be the same unital associative algebra over R). One obtains the value f(r) by substitution of the value r for the symbol X in P. One reason to distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function (see Fermat's little theorem for an example where R is the integers modulo p). This is not the case when R is the real or complex numbers, whence the two concepts are not always distinguished in analysis. An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (like Euclidean division) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for X. And it should be noted that if R is not commutative, there is no (well behaved) notion of polynomial function at all.

Read more about this topic:  Polynomial

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