= Field-valued points do not detect nilpotents
Let $A$ be a <commutative ring> and $I$ an <ideal> all of whose elements are <nilpotent elements>. Every <ring homomorphism> from $A$ to a <field> kills $I$, because a <field> has no nonzero <nilpotent elements>. Thus the <closed immersion> $\operatorname{Spec}(A/I)\hookrightarrow\operatorname{Spec}A$ induces a <bijection> on points valued in every <field>, even when it is not an <isomorphism of schemes>. The <dual numbers> provide the smallest example. Testing the <functor of points> on all <affine schemes> does detect this information, by <affine test schemes determine scheme morphisms>.
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