# Primitive polynomial

From Encyclopedia of Mathematics

A polynomial , where is a unique factorization domain, whose coefficients do not have common factors. Any polynomial can be written in the form with a primitive polynomial and the greatest common divisor of the coefficients of . The element , defined up to multiplication by invertible elements of , is called the content of the polynomial . Gauss' lemma holds: If , then . In particular, a product of primitive polynomials is a primitive polynomial.

#### References

[1] | O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) |

#### Comments

#### References

[a1] | P.M. Cohn, "Algebra" , 1 , Wiley (1982) pp. 165 |

[a2] | G. Birkhoff, S. MacLane, "A survey of modern algebra" , Macmillan (1953) pp. 79 |

**How to Cite This Entry:**

Primitive polynomial.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Primitive_polynomial&oldid=14653

This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article