>>21 補足
Q(S)→Q~(S) or A(S) |Q~ or Aは、Q上の代数的数全体からなる集合は体、とした方が、理論的には綺麗かもね
https://en.wikipedia.org/wiki/Algebraic_number
An algebraic number is a possibly complex number that is a root of a finite, non-zero polynomial in one variable with rational coefficients
(or equivalently ? by clearing denominators ? with integer coefficients). Numbers such as π that are not algebraic are said to be transcendental.

Properties
・The set of algebraic numbers is countable (enumerable).
・Hence, the set of algebraic numbers has Lebesgue measure zero (as a subset of the complex numbers), i.e. "almost all" complex numbers are not algebraic.
・Given an algebraic number, there is a unique monic polynomial (with rational coefficients) of least degree that has the number as a root.
This polynomial is called its minimal polynomial. If its minimal polynomial has degree n, then the algebraic number is said to be of degree n.
An algebraic number of degree 1 is a rational number. A real algebraic number of degree 2 is a quadratic irrational.
・The set of real algebraic numbers is linearly ordered, countable, densely ordered, and without first or last element, so is order-isomorphic to the set of rational numbers.

The field of algebraic numbers
The sum, difference, product and quotient of two algebraic numbers is again algebraic (this fact can be demonstrated using the resultant),
and the algebraic numbers therefore form a field Q~ (sometimes denoted by A, though this usually denotes the adele ring).
Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic. This can be rephrased by saying that the field of algebraic numbers is algebraically closed.
In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.
The set of real algebraic numbers itself forms a field.