polynomial ring over a field (original) (raw)
The theorem implies, similarly as in the ring ℤ of the integers, that one can perform in K[X] 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 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 associates in the ring K[X] and thus d′ is gotten from d by multiplication
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 coprime.
Using the Euclid’s algorithm as in ℤ, 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 product of polynomials in K[X] is divisible by an irreducible polynomial
of K[X], then at least one factor (http://planetmath.org/Product) of the product is divisible by the irreducible polynomial.
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 trinomial X4-X2-2 into monic irreducible prime factors are
(X2-2)(X2+1) in ℚ[X],
(X2-2)(X+i)(X-i) in ℚ(i)[X],
(X+2)(X-2)(X2+1) in ℚ(2)[X],
(X+2)(X-2)(X+i)(X-i) in ℚ(2,i)[X].