# 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.

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#### References

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

#### 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

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.