Algebraic Number Theory is a branch of mathematics that studies the properties of numbers through the lens of algebra. It focuses on algebraic structures, such as rings and fields, to understand the behavior of integers and rational numbers in more complex systems. This field explores concepts like prime numbers, factorization, and number fields, which are extensions of the rational numbers. One of the key goals of Algebraic Number Theory is to solve equations and understand their solutions in different number systems. It also investigates how these solutions relate to algebraic integers, which are roots of polynomial equations with integer coefficients. This area of study has applications in cryptography, coding theory, and other areas of mathematics.