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

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

Let $\Q \sqbrk X$ be the ring of polynomials over $\Q$ in one indeterminate $X$.

Let $\map f X, \map g X \in \Q \sqbrk X$ be primitive polynomials.


Then their product $f g$ is also a primitive polynomial.


Gauss's Lemma on Primitive Polynomials over Ring

Let $R$ be a commutative ring with unity.

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



Then $f g$ is primitive.


Content is Multiplicative

Let $h \in \Q \sqbrk X$ be a polynomial with rational coefficients.

Let $\cont h$ denote the content of $h$.


Then for any polynomials $f, g \in \Q \sqbrk X$ with rational coefficients:

$\cont {f g} = \cont f \cont g$


Statement on irreducible polynomials

Let $\Z$ be the ring of integers.

Let $\Z \sqbrk X$ be the ring of polynomials over $\Z$.

Let $h \in \Z \sqbrk X$ be a polynomial.


The following are equivalent:

$(1): \quad h$ is irreducible in $\Q \sqbrk X$ and primitive
$(2): \quad h$ is irreducible in $\Z \sqbrk X$.


Polynomial ring is UFD

Let $R$ be a unique factorization domain.


Then the ring of polynomials $R \sqbrk X$ is also a unique factorization domain.


Source of Name

This entry was named for Carl Friedrich Gauss.