A result of van der Waerden says Galois theory "needs" incomputable sets - what does this mean, exactly?

galois-theory computability

The gist of the matter is that testing if a polynomial is squarefree can be undecidable in positive characteristic, see below. This has no impact in practice because the bizarre (though computable) fields that are constructed for such counterexamples (e.g. below) do not arise naturally in algebra.

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Above is excerpted from C. F. Miller, Computable Algebra.

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Above is excerpted from von zur Gathen, Hensel and Newton Methods in Valuation Rings, Math. Comp., 1984

Below are the original papers.

M.O. Rabin, Computable Algebra, General Theory, and Theory of Computable Fields, Trans. A.M.S. 95 (1960), 341-360.

A. Frohlich & J.C. Shepherdson. Effective procedures in field theory, Phil. Trans. Royal Soc. London, Series A 248 (1956) 950, 407-432.

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