- A rational number is an algebraic number; a rational number is an algebraic
integer if and only if it is an integer. For clarity, I'll refer to the usual
integers as the
*rational integers*. - For any rational number , the root of unity satisfies the polynomial , and so is an algebraic integer.
- A
*Gaussian integer*, a number of the form , is an algebraic integer. - For any rational number , the numbers and are algebraic integers, and is an algebraic number. Can you explicitly write down polynomials that these are roots of? (These polynomials turn out to have lots of interesting properties.)
- Given a recurrence relation
with
(rational) integer coefficients, all solutions can be expressed in terms of some algebraic
integers. For example, the -th Fibonacci number can be written as
- The eigenvalues of a matrix with (rational) integer entries are algebraic integers. This is one way algebraic numbers come up in topology, group theory, algebraic geometry, combinatorics, etc.

- The set of algebraic numbers is closed under addition, subtraction, multiplication and division. The set of algebraic integers is closed under addition, subtraction and multiplication, but not division.
- The root of a polynomial whose coefficients are algebraic numbers (resp., algebraic integers) is one also.