Algebraic Numbers and Integers

Abstract

In this chapter we introduce the fundamental notions of the theory, and develop some of their properties. Let us start with definitions. Any complex number which is integral over the field ℚ of rational numbers will be called an algebraic number, and if it is also integral over the ring ℤ of rational integers, then it will be called an algebraic integer. Corollary to Proposition 1.6 shows that the set of all algebraic numbers forms a ring, and the same holds for the set of all algebraic integers. Actually the first of these rings is a field, since if a ≠ 0 is algebraic, then it is a root of X ^m + a _m−1 X ^{m − 1} + ... + a _l X + a ₀ with rational a _i ’s and non-zero a ₀, hence a ⁻¹ is a root of the polynomial X ^m + a ₀ ^−l a ₁ X ^m−1 + ... + a ₀ ⁻¹.