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 regular integers and include numbers like rational numbers and irrational numbers. Algebraic integers can be found in various number systems, such as complex numbers and number fields. They play a crucial role in number theory and are used in various mathematical applications, including cryptography and algebraic geometry.