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Abstract Algebra

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Moderate

What does the equation say?

ABSALG-KMEN0O

Let $R=\mathbb{Z}_5[x]$.

Note that $3x^2+3x+4=(x+2)(3x+2)=(x+4)(3x+1)$.

What CANNOT be inferred about $R$ from this equation?

A

Either $R$ is not a UFD or $3x+2$ and $x+4$ are associate.

B

$3x^2+3x+4$ is reducible over $\mathbb{Z}_5$.

C

Either $R$ is not a UFD or $x+2$ and $3x+1$ are associate.

D

$R$ is not a UFD.