  Much is known about the adele ring of an algebraic number field from the
perspective of Harmonic Analysis and Class Field Theory. However, its
ring-theoretical aspects are often ignored. Here we present a description of
the prime spectrum of this ring and study some of the algebraic and topological
properties of these prime ideals. We also study how they behave under separable
extensions of the base field and give an indication of how this study can be
applied in adele rings not of number fields.
