Algebraic number theory

Algebraic number theory is the study of algebraic numbers, which is any complex number which is a root to a polynomial with rational coefficients. One central idea in algebraic number theory is the study of number fields which are algebraic extensions of the rational fields. That is, let $$K$$ be an extension field of $$\mathbb Q$$, the field of rational numbers. We call $$K$$ an algebraic number field if the field extension $$K / \mathbb Q$$ is finite.