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Abstract Algebra

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Ideals in Polynomial Rings Over Finite Fields

ABSALG-LJGFCM

Which of the following are false?

We denote by $(P)$ the ideal generated by $P$ in the ring in question.

Select ALL that apply.

A

$({\mathbb Z}/3{\mathbb Z})[x]/(x^2+1)$ is a field

B

$({\mathbb Z}/5{\mathbb Z}[x])/(x^2+1)$ is a field

C

$(x^2+1)\subseteq (x-2)$ in $({\mathbb Z}/5{\mathbb Z})[x]$

D

$x^2+1$ is reducible in $({\mathbb Z}/15{\mathbb Z})[x]$

E

$(x^2+1)\subseteq (x+2)$ in $({\mathbb Z}/5{\mathbb Z})[x]$