= Solution
A suitable <field with four elements> is
$$
\boxed{R=\mathbb F_2[t]/(t^2+t+1).}
$$
Write $\alpha$ for the class of $t$. <Monic polynomial division over a ring> gives a unique representative $a+b\alpha$, with $a,b\in\mathbb F_2$: existence follows by division, and uniqueness follows because a nonzero multiple of the degree-two modulus cannot have degree less than two. The four elements are therefore $0,1,\alpha,\alpha+1$.
The defining relation gives $\alpha^2=\alpha+1$, and hence $\alpha(\alpha+1)=1$. Thus the nonzero elements have inverses
$$
1^{-1}=1,\qquad\alpha^{-1}=\alpha+1,\qquad(\alpha+1)^{-1}=\alpha.
$$
The <quotient ring> is commutative and every nonzero element is a <unit>, so it is a <field>. This verifies the field property directly, without quoting a theorem about irreducible-polynomial quotients.
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