Primitive Polynomial

Description

We will be very interested in finding out whether a polynomial is reducible or Irreducible polynomial. To do this, the primitive form will be useful.

Definition

Sei $R$ ein faktorieller Ring und $f\in R[x]$ ein Polynom. Wir nennen das Polynom $f$ primitiv, wenn $f\ne 0$ ist und die Koeffizienten $a_0,...,a_n$ keinen gemeinsamen Primteiler besitzen.

Properties

If you want to restrict possible rational roots, this is the tool. It doesn’t say anything on irrational roots however.

Conditions on possible rational roots

Let $R$ be a Unique factorization domain, $K$ its Quotient field and $f\in R[x]$ a polynomial of degree $n\ge 1$. Let $f=a_n{x}^n+...+a_0$.

1. If $\alpha \in K$ is a root of $f$ and $\alpha =\frac{p}{q}$ where $p,q\in R$ coprime, $q\ne 0$ then $q|a_n$ and $p|a_0$.
2. It follows that if $a_n=1$, then $\alpha$ lies in $R$ and is a divisor of $a_0$.

Gauss's Lemma: Preserved under multiplication

If $f,g$ are primitive, then $fg$ as well.