by Ciro Santilli (@cirosantilli, 32)
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.
Open as of 2020:
Video 1. Why π^π^π^π could be an integer by Stand-up Maths (2021) Source. Sponsored by Jane Street. Shame.

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