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No constant polynomials are irreducible?

What does it mean?

Is it because constant polynomials are units?

user88310
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    We need to be more careful with the term "constant polynomial". It could reasonably mean either a degree 0 polynomial, or a polynomial which, as a function of the elements of the ring, is constant. For example, $x^2+x+1\in \mathbb{F}_2[x]$ is irreducible and constantly evaluates to $1,$ so you could call that a constant polynomial that is irreducible. – Ragib Zaman Aug 11 '13 at 04:21

1 Answers1

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You're correct, if working over a field; that is, if the polynomial ring you're talking about is $F[x]$ where $F$ is a field, then any non-zero constant polynomial is a unit of the ring $F[x]$, and the constant zero polynomial is the zero element of the ring $F[x]$, and therefore none of them are irreducible elements of $F[x]$, simply because the definition of "irreducible" excludes such elements.

However, if the polynomial ring is $R[x]$ where $R$ is a ring that is not a field, then there may be non-zero constant polynomials that are irreducible elements of $R[x]$. For example, in the ring $\mathbb{Z}[x]$, the element $2$ is a constant polynomial that is irreducible.

Zev Chonoles
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