This chapter is an introduction to algebraic number fields, which arose from both a generalization of the arithmetic in ℤ and the necessity to solve certain Diophantine equations. After recalling basic concepts from algebra and providing some polynomial irreducibility tools, the ring of integers \(\mathcal {O}_{\mathbb {K}}\) of an algebraic number field \(\mathbb {K}\) is investigated. Next, the ideal numbers, as Kummer called them, are introduced to restore unique factorization. The last section shows how analytic tools can be used to solve hard problems of algebraic number theory. In particular, an account of Zimmert’s method for ideal classes is given.