Page "Algebraic number" Paragraph 18

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, sometimes denoted by A ( which may also denote the adele ring ) or < span style =" text-decoration: overline ;"> Q </ span >.

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.