= Real fourth-power fibre classification
{title2=$\mathbb A^1_{\mathbb R}\to\mathbb A^1_{\mathbb R},\quad x\mapsto x^4$}
The <scheme-theoretic fibres> over positive real points, negative real points, zero, and nonreal <closed points> have real coordinate algebras $\mathbb R^2\times\mathbb C$, $\mathbb C^2$, $\mathbb R[x]/(x^4)$, and $\mathbb C^4$, respectively. Their ordinary <irreducible component> counts are $3,2,1,4$, so there are four real-scheme isomorphism classes. The last case pulls back an irreducible quadratic to a squarefree degree-eight polynomial, with four conjugate pairs of nonreal roots. The zero fibre retains its nilpotents and is not a reduced point.
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