polynomial ring over a field
The degree , defined for every in except the zero polynomial, satisfies the requirements of a Euclidean valuation in . In fact, the degrees of polynomials are non-negative integers. If and belong to and the latter of them is not the zero polynomial, then, as is well known, the long division gives two unique polynomials and in such that
where or 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 a Euclid’s algorithm which yields a greatest common divisor 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 .
Let be a greatest common divisor of certain polynomials. Then apparently also , where is any non-zero element of , is a gcd of the same polynomials. They do not have other gcd’s than , for if is an arbitrary gcd of them, then
Corollary 1. The greatest common divisor of polynomials in the ring is unique up to multiplication by a non-zero element of the field . The monic (http://planetmath.org/Monic2) gcd of polynomials is unique.
If the monic gcd of two polynomials is 1, they may be called coprime.
Using the Euclid’s algorithm as in , one can prove the
Corollary 2. If and are two non-zero polynomials in , this ring contains such polynomials and that
and especially, if and are coprime, then and may be chosen such that .
Corollary 3. If a product of polynomials in is divisible by an irreducible polynomial of , 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 domain.
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 over a field containing the polynomial. Especially, is a UFD.
|Title||polynomial ring over a field|
|Date of creation||2013-03-22 17:42:55|
|Last modified on||2013-03-22 17:42:55|
|Last modified by||pahio (2872)|