As special cases of the theorem of the parent “polynomial equation with algebraic coefficients” of this entry, one obtains the
Corollary. If and are algebraic numbers, then also , ,
and (provided ) are algebraic numbers. If and are algebraic integers, then also , and
are algebraic integers.
The case of needs an additional consideration: If is the minimal polynomial of , the equation implies
Hence is an algebraic number, and therefore also
It follows from the corollary that the set of all algebraic numbers is a field and the set of all algebraic integers is a ring (an integral domain, too). Moreover, the mentioned theorem implies that the field of algebraic numbers is algebraically closed and the ring of algebraic integers integrally closed. The field of algebraic numbers, which is sometimes denoted by , contains for example the complex numbers obtained from rational numbers by using arithmetic operations and taking roots (these numbers form a subfield of ).