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DEFINITION OF A POLYNOMIAL


In Real Analysis , a polynomial is a certain type of a Function of one or several variables (see Polynomial ), or in other words, a ''polynomial function''.

This definition cannot be adapted to a general ring, however. For example, over the ring Z/2Z of Integer s Modulo 2, the polynomial
P

takes only the value 0, as when ''k'' is an integer, ''k''(''k''+1) is always even. But we would expect ''P''(''X'') to be different than the zero polynomial.

The approach taken is then the following. Let ''R'' be a ring. A polynomial ''P''(''X'') is defined to be a formal expression of the form
: P(X) = a_m X^m + a_{m - 1} X^{m - 1} + \cdots + a_1 X + a_0
where the ''coefficients'' ''a''0, ..., ''a''m are elements of the ring ''R'', and ''X'' is considered to be a formal symbol. Two polynomials are considered to be equal if and only if the corresponding coefficients for each power of ''X'' are equal. Polynomials with coefficients in ''R'' can be added by simply adding the corresponding coefficients and multiplied using the Distributive Law , and the rules
: X\, a = a\, X
for all elements ''a'' of the ring ''R'' and
: X^k\, X^l = X^{k+l}
for all Natural Numbers ''k'' and ''l''.


THE POLYNOMIAL RING ''R'' {LINK WITHOUT TITLE}


One can then check that the set of all polynomials with coefficients
in the ring ''R'', together with the addition + and the multiplication \cdot mentioned above, forms itself a ring, the polynomial ring over ''R'', which is denoted by ''R'' {Link without Title} .

Formally these two ring operations are functions defined on R[X] imes R[X] with values in R[X], given by the formulas

:\sum_{i=0}^na_iX^i+
\sum_{i=0}^n b_iX^i=\sum_{i=0}^n(a_i+b_i)X^i

and

:\left(\sum_{i=0}^n a_iX^i ight)\left(\sum_{j=0}^m b_jX^j ight)=\sum_{k=0}^{m+n}\left(\sum_{\mu +
u =k}a_{\mu} b_{
u} ight)X^k.

If ''R'' is Commutative , then ''R'' {Link without Title} is an
Algebra over ''R''.

One can think of the ring ''R'' {Link without Title} as arising from ''R''
by adding one new element ''X'' to ''R'' and only requiring that ''X'' commute with all elements of ''R''. In order for ''R'' {Link without Title} to form a ring, all sums of powers of ''X'' have to be included as well.


THE POLYNOMIAL RING IN SEVERAL VARIABLES


Given two variables ''X'' and ''Y'', one constructs the polynomial ring ''R'' and then, on top of it, the ring (''R''[''X'' ) This ring is considered the polynomial ring in the two variables ''R''[''X'',''Y'' .

For example, the polynomial
:P(X, Y)=X^2Y^2+4XY^2+5X^3-8Y^2+6XY-2Y+7
is thought of as the polynomial
:(X^2+4X-8)Y^2+(6X-2)Y+(5X^3+7)
in ''Y'' with coefficients in ''R'' {Link without Title} .

In similar fashion, the ring ''R'' ..., ''Xn'' in ''n'' variables ''X1'', ..., ''Xn'' is constructed.


Equivalent definition

Polynomials in ''n'' variables can also be defined as functions from N''n'' into ''R'' which are zero everywhere except for a finite number of points, with the addition and ''R''-multiplication defined in the canonical way, and multiplication defined by the Convolution
  • Q : k\mapsto\sum_{i+j=k} P_i\,Q_j ~,

    • where ''i,j,k''∈N''n'' are the (multi-)indices corresponding to respective powers of the indeterminates (and P_i,Q_j are the associated ''coefficients'' of the respective polynomial).


The link with the traditional notation is made by writing as X_p^q the elements of the Canonical Basis of this Free Module , which are the functions associating to a vector (0...0,q,0...0) of N''n'' the value 1''R'', and zero to any other vector of N''n'' (where ''q'' is in the ''p''-th place of the vector).


PROPERTIES




SOME USES OF POLYNOMIAL RINGS

Factoring out Ideals from a polynomial ring is an important tool for constructing new rings out of known ones.

For instance, the clean construction of Finite Field s involves the use of those operations, starting out with the field of integers modulo some Prime Number as the coefficient ring ''R'' (see Modular Arithmetic ).

An interesting example of a ring obtained by using polynomials is the ring of Frobenius Polynomial s, where the ring multiplication is given by function composition, rather than by polynomial multiplication.

The polynomial ring can be used to classify all Simple Field Extension .