Divisibility Polynomial



in commutative algebra, 1 major focus of study divisibility among polynomials. if r integral domain , f , g polynomials in r[x], said f divides g or f divisor of g if there exists polynomial q in r[x] such f q = g. 1 can show every 0 gives rise linear divisor, or more formally, if f polynomial in r[x] , r element of r such f(r) = 0, polynomial (x − r) divides f. converse true. quotient can computed using polynomial long division.


if f field , f , g polynomials in f[x] g ≠ 0, there exist unique polynomials q , r in f[x] with







f
=
q

g
+
r


{\displaystyle f=q\,g+r}



and such degree of r smaller degree of g (using convention polynomial 0 has negative degree). polynomials q , r uniquely determined f , g. called euclidean division, division remainder or polynomial long division , shows ring f[x] euclidean domain.


analogously, prime polynomials (more correctly, irreducible polynomials) can defined non-zero polynomials cannot factorized product of 2 non-constant polynomials. in case of coefficients in ring, non-constant must replaced non-constant or non-unit (both definitions agree in case of coefficients in field). polynomial may decomposed product of invertible constant product of irreducible polynomials. if coefficients belong field or unique factorization domain decomposition unique order of factors , multiplication of non-unit factor unit (and division of unit factor same unit). when coefficients belong integers, rational numbers or finite field, there algorithms test irreducibility , compute factorization irreducible polynomials (see factorization of polynomials). these algorithms not practicable hand-written computation, available in computer algebra system. eisenstein s criterion can used in cases determine irreducibility.








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