  We show that the theory of hyperrings, due to M. Krasner, supplies a perfect
framework to understand the algebraic structure of the adele class space of a
global field. After promoting F1 to a hyperfield K, we prove that a hyperring
of the form R/G (where R is a ring and G is a subgroup of its multiplicative
group) is a hyperring extension of K if and only if G union 0 is a subfield of
R. This result applies to the adele class space which thus inherits the
structure of a hyperring extension H of K. We begin to investigate the content
of an algebraic geometry over K. The category of commutative hyperring
extensions of K is inclusive of: commutative algebras over fields with
semi-linear homomorphisms, abelian groups with injective homomorphisms and a
rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we
show that for a global field of positive characteristic, the groupoid of the
prime elements of the hyperring H is canonically and equivariantly isomorphic
to the groupoid of the loops of the maximal abelian cover of the curve
associated to the global field.
