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I read the following definition of polynomial reudcibility:

Let $A$ be one of the sets $\mathbb{Z}, \mathbb{Q}, \mathbb{R}$ and $\mathbb{C}$. Let $f\in A[X]$ be a polynomial of degree $n\in \mathbb{N}$. We say that $f$ is reducible over $A$ if $\exists g, h \in A[X]$, $\operatorname{deg}g, \operatorname{deg}h<n$, such that $f=g h$.

This definition puzzles me. I know that $f=2X+2=2(X+1)\in \mathbb{Z}[X]$ is reducible, but, correct me if I am wrong, it should be irreducible according to this definition.

ChemistryGeek
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1 Answers1

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You are right, in this definition it is (probably) assumed that $g,h$ then have positive degree, see this post for a similar question.

So the correct definition should be as follows.

Definition: An element $f$ in a commutative ring $R$ with $1$ is called irreducible if it is non-zero, not a unit and if from each representation $f=gh$ with $g,h\in R$ always follows that $g$ or $h$ is a unit. Otherwise $f$ is called reducible.

Example: Let $R=\Bbb Z[X]$ and $f=2X+2$. Then $f$ is reducible, since $2$ is not a unit in $R$. Note that $R^{\times}=\Bbb Z^{\times}=\{\pm 1\}$.

Dietrich Burde
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