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Abstract Algebra

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Moderate

Irreducible Polynomial Over the Integers (definition)

ABSALG-1LVECD

Which of the following are equivalent to a non-constant polynomial $P(x)\in {\mathbb Z}[x]$ being irreducible over ${\mathbb Z}$, that is, being irreducible as an element of ${\mathbb Z}[x]$?

Select ALL that apply.

A

The coefficients of $P(x)$ have greatest common divisor equal $1$

B

If there are polynomials $Q(x)$ and $R(x)$ in ${\mathbb Z}[x]$ such that $P(x)=Q(x)R(x)$, then one of $Q(x)$ or $R(x)$ has degree equal $0$

C

The polynomial $P(x)$ is primitive and if $P(x)=Q(x)R(x)$ with $Q(x)$, $R(x)$ in ${\mathbb Z}[x]$, then one of $Q(x)$, $R(x)$ has degree equal $0$

D

The polynomial $P(x)$ is primitive and is irreducible over ${\mathbb Q}$ (namely, is irreducible as an element of ${\mathbb Q}[x]$)

E

None of the above is equivalent to $P(x)$ being irreducible in ${\mathbb Z}[x]$