Incoming links: Cardinality
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.
Chuck Norris counted to infinity. Twice.
There are a few related concepts that are called infinity in mathematics:
- limits that are greater than any number
- the cardinality of a set that does not have a finite number of elements
- in some number systems, there is an explicit "element at infinity" that is not a limit, e.g. projective geometry
The order of a algebraic structure is just its cardinality.