Let p and q be polynomials with coefficients in an integral domain F, typically a field or the integers. A greatest common divisor of p and q is a polynomial d that divides p and q and such that every common divisor of p and q also divides d. Every pair of polynomials has a GCD if and only if F is a unique factorization domain.
If F is a field and p and q are not both zero, d is a greatest common divisor if and only if it divides both p and q and it has the greatest degree among the polynomials having this property. If p = q = 0, the GCD is 0. However, some authors consider that it is not defined in this case.
The greatest common divisor of p and q is usually denoted "gcd(p, q)".
The greatest common divisor is not unique: if d is a GCD of p and q, then the polynomial f is another GCD if and only if there is an invertible element u of F such that
In other words, the GCD is unique up to the multiplication by an invertible constant.
In the case of the integers, this indetermination has been settled by choosing, as the GCD, the unique one which is positive (there is another one, which is its opposite). With this convention, the GCD of two integers is also the greatest (for the usual ordering) common divisor. When one want to settle this indetermination in the polynomial case, one lacks of a natural total order. Therefore, one chooses once for all a particular GCD that is then called the greatest common divisor. For univariate polynomials over a field, this is usually the unique GCD which is monic (that is has 1 as coefficient of the highest degree). In more general cases, there is no general convention and above indetermination is usually kept. Therefore equalities like d = gcd(p, q) or gcd(p, q) = gcd(r, s) are usual abuses of notation which should be read "d is a GCD of p and q" and "p, q has the same set of GCD as r, s". In particular, gcd(p, q) = 1 means that the invertible constants are the only common divisors, and thus that p and q are coprime.
Read more about this topic: Greatest Common Divisor Of Two Polynomials
Other articles related to "general definition, definitions, definition, general":
... One argument is that in most subparts and the general definition, the Internal Revenue Code's definitions of "state" and "United States," are what other amending code sections refer to as "a special definition of "state ... Under this argument, the definition of "state" within the Code in general, or within those certain subparts of the Code, refers only to territories, or the ... Alaska and Hawaii were formerly included in the "special" definition of a "state" until each was removed from the that general definition by the Alaska Omnibus Act ...
... The definition of combinatorial map in any dimension is given in and An n-dimensional combinatorial map (or n-map) is a (n + 1)-tuple M = (D, β1.. ... The last line of this definition fixes constraints which guarantee the topological validity of the represented object a combinatorial map represents ... The initial definition of 2-dimensional combinatorial maps can be retrieved by fixing n = 2 and renaming σ by β1 and α by β2 ...
... In general, an approximate symmetry arises when there are very strong interactions that obey that symmetry, along with weaker interactions that do not ...
Famous quotes containing the words definition and/or general:
“Im beginning to think that the proper definition of Man is an animal that writes letters.”
—Lewis Carroll [Charles Lutwidge Dodgson] (18321898)
“Can a woman become a genius of the first class? Nobody can know unless women in general shall have equal opportunity with men in education, in vocational choice, and in social welcome of their best intellectual work for a number of generations.”
—Anna Garlin Spencer (18511931)