# Gauss’s lemma II

Definition. A polynomial^{} $P={a}_{n}{x}^{n}+\mathrm{\cdots}+{a}_{0}$ over an integral domain^{}
$D$ is said to be *primitive* if its coefficients are not all divisible
by any element of $D$ other than a unit.

Proposition^{} (Gauss). Let $D$ be a unique factorization domain^{} and $F$ its field of fractions^{}.
If a polynomial $P\in D[x]$ is reducible in $F[x]$, then it is reducible in $D[x]$.

Remark. The above statement is often used in its contrapositive form. For an example of this usage, see this entry (http://planetmath.org/AlternativeProofThatSqrt2IsIrrational).

*Proof.* A primitive polynomial in $D[x]$ is by definition divisible by a non invertible^{} constant polynomial, and therefore reducible in $D[x]$ (unless it is itself constant). There is therefore nothing to prove unless $P$ (which is not constant) is primitive. By assumption^{} there exist non-constant $S,T\in F[x]$ such that $P=ST$. There are elements $a,b\in F$ such that $aS$ and $bT$ are in $D[x]$ and are primitive (first multiply by a nonzero element of $D$ to chase any denominators, then divide by the gcd of the resulting coefficients in $D$). Then $aSbT=abP$ is primitive by Gauss’s lemma I, but $P$ is primitive as well, so $ab$ is a unit of $D$ and $P={(ab)}^{-1}(aS)(bT)$ is a nontrivial decomposition of $P$ in $D[X]$. This completes^{} the proof.

Remark. Another result with the same name is Gauss’ lemma on quadratic residues^{}.

From the above proposition and its proof one may infer the

Theorem. If a primitive polynomial of $D[x]$ is divisible in $F[x]$, then it splits in $D[x]$ into primitive prime factors. These are uniquely determined up to unit factors of $D$.

Title | Gauss’s lemma II |

Canonical name | GausssLemmaII |

Date of creation | 2013-03-22 13:07:52 |

Last modified on | 2013-03-22 13:07:52 |

Owner | bshanks (153) |

Last modified by | bshanks (153) |

Numerical id | 18 |

Author | bshanks (153) |

Entry type | Theorem |

Classification | msc 12E05 |

Synonym | Gauss’ lemma II |

Related topic | GausssLemmaI |

Related topic | EisensteinCriterion |

Related topic | ProofOfEisensteinCriterion |

Related topic | PrimeFactorsOfXn1 |

Related topic | AlternativeProofThatSqrt2IsIrrational |

Defines | primitive polynomial |