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This is Cipolla's algorithm, I believe. The integers mod $p$ we call $\mathbf{F}_p$, and since $h^2 -4x$ is a quadratic nonresidue mod $p$, the polynomial $P(Y) := Y^2-(h^2-4x) \in \mathbf{F}_p[Y]$ has no roots. We can then mod out the polynomial ring $\mathbf{F}_p[Y]$ to get $\mathbf{K} := \mathbf{F}_p[Y]/(Y^2-(h^2-4x))$ (which by some algebra we know is ...

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