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
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