The set of all algebraic numbers forms a field.

This field contains all of the rational numbers, but it is a quadratically closed field.

Like the rationals, this field also has the same cardinality as the natural numbers, because we can specify and enumerate each of its members by a fixed number of integers from the polynomial equation that defines them. So it is a bit like the rationals, but we use potentially arbitrary numbers of integers to specify each number (polynomial coefficients + index of which root we are talking about) instead of just always two as for the rationals.

Each algebraic number also has a degree associated to it, i.e. the degree of the polynomial used to define it.

TODO understand.

Sometimes mathematicians go a little overboard with their naming.

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There's a billion simple looking expressions which are not known to be transcendental numbers or not. It's cute simple to state but hard to prove at its best.

Open as of 2020:

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Polynomial (possibly a multivariate polynomial) with integer coefficients.

Sometimes systems of Diophantine equations are considered.

Problems generally involve finding integer solutions to the equations, notably determining if any solution exists, and if infinitely solutions exist.

The general problem is known to be undecidable: Hilbert's tenth problem.

The Pythagorean triples, and its generalization Fermat's last theorem, are the quintessential examples.

Direct consequence of Euclid's formula.

A generalization of the Pythagorean triple infinity question.

Once you hear about the uncomputability of such problems, it makes you see that all Diophantine equation questions risk being undecidable, though in some simpler cases we manage to come up with answers. The feeling is similar to watching people trying to solve the Halting problem, e.g. in the effort to determine BB(5).

mathoverflow.net/questions/11540/what-are-the-most-attractive-turing-undecidable-problems-in-mathematics/103415#103415 provides a specific single undecidable Diophantine equation.

And when it can't, attempt to classify which subset of the integers can be reached. E.g. Legendre's three-square theorem.