An algebraic number field F is a finite extension field of the rational numbers ℚ. It can be generated by a root p of a monic irreducible polynomial $$f(t) = {{t}^{n}} + {{a}_{1}}{{t}^{{n - 1}}} + {\text{ }} \ldots + {{a}_{n}}\epsilon \mathbb{Z}[t]$$ , (27) where n is also called the degree of F. Clearly, ℚ(p) = F ≅ ℚ[t]/f(t)ℚ[t], and the successive powers l, p,…, p n-1 form a basis of F over ℚ. For describing the arithmetic in F we will need the counterpart of the rational integers in F. These integers of F are defined as those elements of F which are algebraic integers, i.e. zeros of monic non-constant polynomials of ℤ[t]. From (27) we conclude that p itself is an integer of F. We proceed to show that the integers of F form a ring.