Gauss's Lemma (Polynomial Theory)

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Theorem

Gauss's lemma on polynomials may refer to any of the following statements.


Product of primitive polynomials is primitive

Rational Polynomial

Let $\Q$ be the field of rational numbers.

Let $\Q \left[{X}\right]$ be the ring of polynomials over $\Q$ in one indeterminate $X$.

Let $f \left({X}\right), g \left({X}\right) \in \Q \left[{X}\right]$ be primitive polynomials.


Then their product $fg$ is also a primitive polynomial.


General Ring

Let $R$ be a commutative ring with unity.

Let $f,g \in R[X]$ be primitive polynomials.


Then $fg$ is primitive.


Content is multiplicative

Rational polynomials

Let $h \in \Q \left[{X}\right]$ be a polynomial with rational coefficients.

Let $c_h = \operatorname{cont} \left({h}\right)$ denote the content of $h$.


Then for any polynomials $f, g \in \Q \left[{X}\right]$ with rational coefficients:

$\operatorname{cont} \left({f g}\right) = \operatorname{cont} \left({f}\right) \operatorname{cont} \left({g}\right)$

Dedekind domain

Let $R$ be a Dedekind domain.

Let $f, g \in R[X]$ be polynomials.

Let $\operatorname{cont} (f)$ denote the content of $f$.


Then $\operatorname{cont} (fg) = \operatorname{cont} (f) \operatorname{cont} (g)$ is the product of $\operatorname{cont}(f)$ and $\operatorname{cont}(g)$.


Statement on irreducible polynomials

Let $\Z$ be the ring of integers.

Let $\Z \left[{X}\right]$ be the ring of polynomials over $\Z$.

Let $h \in \Z \left[{X}\right]$ be a polynomial.


The following are equivalent:

$(1):\quad$ $h$ is irreducible in $\Q \left[{X}\right]$ and primitive
$(2):\quad$ $h$ is irreducible in $\Z \left[{X}\right]$.


Polynomial ring is UFD

Let $R$ be a unique factorization domain.


Then the ring of polynomials $R \left[{X}\right]$ is also a unique factorization domain.


Source of Name

This entry was named for Carl Friedrich Gauss.