Editing Polynomial (section)

== Polynomial ring ==
{{Main|Polynomial ring}}
A ''polynomial'' {{math|''f''}} over a [[commutative ring]] {{math|''R''}} is a polynomial all of whose coefficients belong to {{math|''R''}}. It is straightforward to verify that the polynomials in a given set of indeterminates over {{math|''R''}} form a commutative ring, called the ''polynomial ring'' in these indeterminates, denoted <math>R[x]</math> in the univariate case and <math>R[x_1,\ldots, x_n]</math> in the multivariate case.

One has 
<math display="block">R[x_1,\ldots, x_n]=\left(R[x_1,\ldots, x_{n-1}]\right)[x_n].</math>
So, most of the theory of the multivariate case can be reduced to an iterated univariate case.

The map from {{math|''R''}} to {{math|''R''[''x'']}} sending {{math|''r''}} to itself considered as a constant polynomial is an injective [[ring homomorphism]], by which {{math|''R''}} is viewed as a subring of {{math|''R''[''x'']}}. In particular, {{math|''R''[''x'']}} is an [[algebra (ring theory)|algebra]] over {{math|''R''}}.

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

Formation of the polynomial ring, together with forming factor rings by factoring out [[ideal (ring theory)|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 {{math|''R''[''x'']}} over the real numbers by factoring out the ideal of multiples of the polynomial {{math|''x''<sup>2</sup> + 1}}. Another example is the construction of [[finite field]]s, which proceeds similarly, starting out with the field of integers modulo some [[prime number]] as the coefficient ring {{math|''R''}} (see [[modular arithmetic]]).

If {{math|''R''}} is commutative, then one can associate with every polynomial {{math|''P''}} in {{math|''R''[''x'']}} a ''polynomial function'' {{math|''f''}} with domain and range equal to {{math|''R''}}.  (More generally, one can take domain and range to be any same [[unital algebra|unital]] [[associative algebra]] over {{math|''R''}}.)  One obtains the value {{math|''f''(''r'')}} by [[substitution (algebra)|substitution]] of the value {{math|''r''}} for the symbol {{math|''x''}} in {{math|''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 {{math|''R''}} is the integers modulo {{math|''p''}}). This is not the case when {{math|''R''}} is the real or complex numbers, whence the two concepts are not always distinguished in [[analysis (mathematics)|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 {{math|''x''}}.

=== Divisibility ===
{{Main|Polynomial greatest common divisor|Factorization of polynomials}}
If {{math|''R''}} is an [[integral domain]] and {{math|''f''}} and {{math|''g''}} are polynomials in {{math|''R''[''x'']}}, it is said that {{math|''f''}} ''divides'' {{math|''g''}} or {{math|''f''}} is a divisor of {{math|''g''}} if there exists a polynomial {{math|''q''}} in {{math|''R''[''x'']}} such that {{math|''f'' ''q'' {{=}} ''g''}}. If <math>a\in R,</math> then {{mvar|a}} is a root of {{mvar|f}} if and only <math>x-a</math> divides {{mvar|f}}. In this case, the quotient can be computed using the [[polynomial long division]].<ref>{{Cite book |last=Irving |first=Ronald S. |title=Integers, Polynomials, and Rings: A Course in Algebra |publisher=Springer |year=2004 |isbn=978-0-387-20172-6 |page=129 |url=https://books.google.com/books?id=B4k6ltaxm5YC&pg=PA129}}</ref><ref>{{cite book |last=Jackson |first=Terrence H. |title=From Polynomials to Sums of Squares |publisher=CRC Press |year=1995 |isbn=978-0-7503-0329-3 |page=143 |url=https://books.google.com/books?id=LCEOri2-doMC&pg=PA143}}</ref>

If {{math|''F''}} is a [[field (mathematics)|field]] and {{math|''f''}} and {{math|''g''}} are polynomials in {{math|''F''[''x'']}} with {{math|''g'' ≠ 0}}, then there exist unique polynomials {{math|''q''}} and {{math|''r''}} in {{math|''F''[''x'']}} with
<math display="block"> f = q \, g + r </math>
and such that the degree of {{math|''r''}} is smaller than the degree of {{math|''g''}} (using the convention that the polynomial 0 has a negative degree). The polynomials {{math|''q''}} and {{math|''r''}} are uniquely determined by {{math|''f''}} and {{math|''g''}}. This is called ''[[Euclidean division of polynomials|Euclidean division]], division with remainder'' or ''polynomial long division'' and shows that the ring {{math|''F''[''x'']}} is a [[Euclidean domain]].

Analogously, ''prime polynomials'' (more correctly, ''[[irreducible polynomial]]s'') can be defined as ''non-zero polynomials which cannot be factorized into the product of two non-constant polynomials''. In the case of coefficients in a ring, ''"non-constant"'' must be replaced by ''"non-constant or non-[[unit (ring theory)|unit]]"'' (both definitions agree in the case of coefficients in a field). Any polynomial may be decomposed into the product of an invertible constant by a product of irreducible polynomials. If the coefficients belong to a field or a [[unique factorization domain]] this decomposition is unique up to the order of the factors and the multiplication of any non-unit factor by a unit (and division of the unit factor by the same unit). When the coefficients belong to integers, rational numbers or a finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (see ''[[Factorization of polynomials]]''). These algorithms are not practicable for hand-written computation, but are available in any [[computer algebra system]]. [[Eisenstein's criterion]] can also be used in some cases to determine irreducibility.