Algebraic integers are a special type of number that are roots of polynomial equations with integer coefficients. Specifically, they are solutions to equations of the form x^n + a_n-1x^n-1 + \ldots + a_1x + a_0 = 0, where the coefficients a_i are integers. An example of an algebraic integer is the square root of 2, which is a solution to the equation x^2 - 2 = 0. These numbers extend the concept of integers beyond the usual whole numbers. They include not only rational numbers but also certain irrational numbers, such as the roots of polynomials. Algebraic integers play a crucial role in number theory and are closely related to concepts like rings and fields in mathematics.