Primitive part and content

In algebra, the content of a polynomial with integer coefficients is the greatest common factor of its coefficients. Thus, e.g., the content of ${\displaystyle 12x^{3}+30x-20}$ equals 2, since this is the greatest common factor of 12, 30, and -20. The definition may be extended to polynomials with coefficients in any fixed unique factorization ___domain.

A polynomial is primitive if it has content unity.

Gauss's lemma for polynomials states that the product of primitive polynomials (with coefficients in the same unique factorization ___domain) also is primitive. Equivalently, it may be expressed as stating that the content of the product of two polynomials is the product of their contents.

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.