Primitive part and content

In algebra, the content of a polynomial with integer coefficients (or, more generally, with coefficients in a unique factorization domain) is the greatest common divisor of its coefficients. The primitive part of such a polynomial is the quotient of the polynomial by its content. Thus a polynomial is the product of its primitive part and its content, and this factorization is unique up to the multiplication of the content by a unit of the ring of the coefficients (and the multiplication of the primitive part by the inverse of the unit).

A polynomial is primitive if its content equals 1. Thus the primitive part of a polynomial is a primitive polynomial.

Gauss's lemma for polynomials states that the product of primitive polynomials (with coefficients in the same unique factorization domain) also is primitive. This implies that the content and the primitive part of the product of two polynomials are, respectively, the product of the contents and the product of the primitive parts.

As the computation of greatest common divisors is generally much easier than polynomial factorization, the first step of a polynomial factorization algorithm is generally the computation of its primitive part–content factorization (see Factorization of polynomials § Primitive part–content factorization). Then the factorization problem is reduced to factorize separately the content and the primitive part.

Content and primitive part may be generalized to polynomials over the rational numbers, and, more generally, to polynomials over the field of fractions of a unique factorization domain. This makes essentially equivalent the problems of computing greatest common divisors and factorization of polynomials over the integers and of polynomials over the rational numbers.

Over the integers

For a polynomial with integer coefficients, the content may be either the greatest common divisor of the coefficients of its additive inverse. The choice is somehow arbitrary, and may depend on a further convention, which is commonly that the leading coefficient of the primitive part be positive.

For example, the content of $-12x^{3}+30x-20$ may be either 2 or –2, since 2 is the greatest common divisor of –12, 30, and -20. If one chooses 2 as the content, the primitive part of this polynomial is

$-6x^{3}+15x-10={\frac {-12x^{3}+30x-20}{2}},$

and thus the primitive–part–content factorization is

$-12x^{3}+30x-20=2(-6x^{3}+15x-10).$

For aesthetic reasons, one often prefer choosing a negative content, here –2, for having the primitive–part–content factorization

$-12x^{3}+30x-20=-2(6x^{3}-15x+10).$

Properties

In the remaining of this article, we consider polynomials over a unique factorization domain R, which can typically be the ring of integers, or a polynomial ring over a field. In R, greatest common divisors are well defined, and are unique up to the multiplication by a unit of R.

The content c(P) of a polynomial P with coefficients in R is the greatest common divisor of its coefficients, and, as such, is defined up to the multiplication by a unit. The primitive part pp(P) of P is the quotient P/c(P) of P by its content; it is a polynomial with coefficients in R, which is unique up to the multiplication by a unit. If the content is changed by multiplication by a unit u, then the primitive part must be changed by dividing it by the same unit, in order to keep the equality

$P=c(P)\operatorname {pp} (P),$

which is called the prime-part-content factorization of P.

The main properties of the content and the primitive part result of Gauss's lemma, which asserts that the product of two primitive polynomials is primitive, where a polynomial is primitive if 1 is a greatest common divisor of its coefficients. This implies:

• The content of a product of polynomial is the product of their contents:

$c(P_{1}P_{2})=c(P_{1})c(P_{2})$

• The primitive part of a product of polynomials is the product of their primitive parts:

$\operatorname {pp} (P_{1}P_{2})=\operatorname {pp} (P_{1})\operatorname {pp} (P_{2})$

• The content of a greatest common divisor of polynomials is the greatest common divisor (in R) of their contents:

$c(\operatorname {gcd} (P_{1},P_{2}))=\operatorname {gcd} (c(P_{1}),c(P_{2}))$

• The primitive part of a greatest common divisor of polynomials is the greatest common divisor (in R) of their primitive parts:

$\operatorname {pp} (\operatorname {gcd} (P_{1},P_{2}))=\operatorname {gcd} (\operatorname {pp} (P_{1}),\operatorname {pp} (P_{2}))$

• The complete factorization of a polynomial over R is the product of the factorization (in R) of the content and of the factorization (in the polynomial ring) of the primitive part.

The last property implies that the computation of the prime-part-content factorization of a polynomial reduces the computation of its complete factorization to the separate factorization of the content and the primitive part. This is generally interesting, because the computation of the prime-part-content factorization involves only greatest common divisor computation in R, which is usually much easier than factorization.

See also

• Rational root theorem

References

• B. Hartley; T.O. Hawkes (1970). Rings, modules and linear algebra. Chapman and Hall. ISBN 0-412-09810-5.
• Page 181 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001
• David Sharpe (1987). Rings and factorization. Cambridge University Press. pp. 68–69. ISBN 0-521-33718-6.