Is $\mathbb{Z}[x]/\langle 4,x^2+x+1 \rangle$ a field?

Hint $ $ Let $\,h = x^2\!+\!x\!+\!1.\ $ If $\,2\,$ is a unit: $\,2 f = 1 + 4g + hh'\,$ in $\Bbb Z[x]\,$ $\overset{\bmod 2}\Longrightarrow\,h\mid 1\,$ in $\,\Bbb Z_2[x]\ \Rightarrow\!\Leftarrow$

And if $\,2=0\,$ then $\,2 = 4g + hh'\,$ so $\,2\mid h'\,$ hence $\,1 = 2g+ h(h'/2)\,$ $\overset{\bmod 2}\Longrightarrow\, h\mid 1\,$ in $\Bbb Z_2[x]\ \Rightarrow\!\Leftarrow$

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Abstract Algebra

Ring Theory

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