polynomial ring over a field

Theorem.  The polynomial ringMathworldPlanetmath over a field is a Euclidean domainMathworldPlanetmath.

Proof.  Let K[X] be the polynomial ring over a field K in the indeterminate X.  Since K is an integral domainMathworldPlanetmath and any polynomial ring over integral domain is an integral domain, the ring K[X] is an integral domain.

The degree ν(f), defined for every f in K[X] except the zero polynomialMathworldPlanetmathPlanetmath, satisfies the requirements of a Euclidean valuation in K[X].  In fact, the degrees of polynomials are non-negative integers.  If f and g belong to K[X] and the latter of them is not the zero polynomial, then, as is well known, the long divisionf/g  gives two unique polynomialsMathworldPlanetmathPlanetmath q and r in K[X] such that


where  ν(r)<ν(g)  or  r is the zero polynomial.  The second property usually required for the Euclidean valuation, is justified by


The theorem implies, similarly as in the ring of the integers, that one can perform in K[X] a Euclid’s algorithm which yields a greatest common divisorMathworldPlanetmathPlanetmath of two polynomials.  Performing several Euclid’s algorithms one obtains a gcd of many polynomials; such a gcd is always in the same polynomial ring K[X].

Let d be a greatest common divisor of certain polynomials.  Then apparently also kd, where k is any non-zero element of K, is a gcd of the same polynomials.  They do not have other gcd’s than kd, for if d is an arbitrary gcd of them, then


i.e. d and d are associatesMathworldPlanetmath in the ring K[X] and thus d is gotten from d by multiplicationPlanetmathPlanetmath by an element of the field K.  So we can write the

Corollary 1.  The greatest common divisor of polynomials in the ring K[X] is unique up to multiplication by a non-zero element of the field K. The monic (http://planetmath.org/Monic2) gcd of polynomials is unique.

If the monic gcd of two polynomials is 1, they may be called coprimeMathworldPlanetmathPlanetmath.

Using the Euclid’s algorithm as in , one can prove the

Corollary 2.  If f and g are two non-zero polynomials in K[X], this ring contains such polynomials u and v that


and especially, if f and g are coprime, then u and v may be chosen such that  uf+vg=1.

Corollary 3.  If a productMathworldPlanetmath of polynomials in K[X] is divisible by an irreducible polynomialMathworldPlanetmath of K[X], then at least one factor (http://planetmath.org/Product) of the product is divisible by the irreducible polynomial.

Corollary 4.  A polynomial ring over a field is always a principal ideal domainMathworldPlanetmath.

Corollary 5.  The factorisation of a non-zero polynomial, i.e. the of the polynomial as product of irreducible polynomials, is unique up to constant factors in each polynomial ring K[X] over a field K containing the polynomial.  Especially, K[X] is a UFD.

Example.  The factorisations of the trinomialX4-X2-2  into monic irreducible prime factorsMathworldPlanetmathPlanetmath are
(X2-2)(X2+1)  in  [X],
(X2-2)(X+i)(X-i)  in  (i)[X],
(X+2)(X-2)(X2+1)  in  (2)[X],
(X+2)(X-2)(X+i)(X-i)  in  (2,i)[X].

Title polynomial ring over a field
Canonical name PolynomialRingOverAField
Date of creation 2013-03-22 17:42:55
Last modified on 2013-03-22 17:42:55
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Theorem
Classification msc 13F07
Related topic FieldAdjunction
Related topic PolynomialRingOverIntegralDomain
Related topic PolynomialRingWhichIsPID
Defines coprime