# Primitive polynomial

A polynomial $f(X) \in R[X]$, where $R$ is a unique factorization domain, whose coefficients do not have common factors. Any polynomial $g(X) \in R[X]$ can be written in the form $g(X) = c(g) f(X)$ with $f(X)$ a primitive polynomial and $c(g)$ the greatest common divisor of the coefficients of $g(X)$. The element $c(G) \in R$, defined up to multiplication by invertible elements of $R$, is called the content of the polynomial $g(X)$. Gauss' lemma holds: If $g_1(X), g_2(X) \in R[X]$, then $c(g_1g_2) = c(g_1)c(g_2)$. 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 |

#### Comments

In the theory of finite, or Galois fields, a *primitive polynomial* is a polynomial $f$ over a finite field $F$ whose roots are primitive elements, in the sense that each is a generator of the cyclic group $E^*$ where $E$ is the extension of $F$ by the roots of $f$, cf Galois field structure.

#### References

[b1] | R. Lidl, H. Niederreiter, "Finite fields" , Addison-Wesley (1983); second edition Cambridge University Press (1996) Zbl 0866.11069 |

**How to Cite This Entry:**

Primitive polynomial.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Primitive_polynomial&oldid=34241