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# factorization of primitive polynomial

As an application of the parent entry we take the factorization of a primitive polynomial of $\mathbb{Z}[x]$ into primitive prime factors. We shall see that the procedure may be done by performing a finite number of tests.

Let

$a(x)\;=:\;a_{n}x^{n}\!+\!a_{{n-1}}x^{{n-1}}\!+\ldots+\!a_{0}$ |

be a primitive polynomial in $\mathbb{Z}[x]$.

By the rational root theorem and the factor theorem, one finds all first-degree prime factors $x\!-\!a$ and thus all primitive prime factors of the polynomial $a(x)$.

If $a(x)$ has a primitive quadratic factor, then it has also a factor

$\displaystyle x^{2}\!+\!px\!+\!q$ | (1) |

where $p$ and $q$ are rationals (and conversely). For settling the existence of such a factor we treat $p$ and $q$ as unknowns and perform the long division

$a(x):(x^{2}\!+\!px\!+\!q).$ |

It gives finally the remainder $b(p,\,q)\,x+c(p,\,q)$ where $b(p,\,q)$ and $c(p,\,q)$ belong to $\mathbb{Z}[p,\,q]$. According to the parent entry we bring the system

$\displaystyle\begin{cases}b(p,\,q)\;=\;0\\ c(p,\,q)\;=\;0\end{cases}$ |

to the form

$\displaystyle\begin{cases}\bar{b}(q)\;=\;0\\ \bar{c}(p,\,q)\;=\;0\end{cases}$ |

and then can determine the possible rational solutions $(p,\,q)$ of the system via a finite number of tests. Hence we find the possible quadratic factors (1) having rational coefficients. Such a factor is converted into a primitive one when it is multiplied by the gcd of the denominators of $p$ and $q$.

Determining a possible cubic factor $x^{3}\!+\!px^{2}\!+\!qx\!+\!r$ with rational coefficients requires examination of a remainder of the form

$b(p,\,q,\,r)\,x^{2}+c(p,\,q,\,r)\,x+d(p,\,q,\,r).$ |

In the needed system

$\displaystyle\begin{cases}b(p,\,q,\,r)\;=\;0\\ c(p,\,q,\,r)\;=\;0\\ d(p,\,q,\,r)\;=\;0\end{cases}$ |

we have to perform two eliminations. Then we can act as above and find a primitive cubic factor of $a(x)$. Similarly also the primitive factors of higher degree. All in all, one needs only look for factors of degree $\leq\frac{n}{2}$.

# References

- 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).

## Mathematics Subject Classification

12D99*no label found*26C05

*no label found*

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