algebraic integer

An algebraic integer is a special kind of number that solves a polynomial equation with whole number coefficients, where the solutions are contained within a number system extending ordinary integers. In simpler terms, it's a number that, when plugged into a polynomial with whole numbers, makes the equation true, and stays within a particular mathematical universe that includes numbers like square roots or cube roots. Algebraic integers generalize ordinary integers (like 1, 2, or -3) and are fundamental in advanced number theory, especially in understanding solutions to equations and the structure of number systems beyond the counting numbers.