Solutions of the quartic Fermat equation in ring class fields of odd conductor over quadratic fields K=Q(\sqrt{-d}) with -d ≡ 1 (mod 8) are shown to be periodic points of a fixed algebraic function T(z) defined on the punctured disk 0< |z|₂ ≦ (1/2) of the maximal unramified, algebraic extension K₂ of the 2-adic field Q₂. All ring class fields of odd conductor over imaginary quadratic fields in which the prime p=2 splits are shown to be generated by complex periodic points of the algebraic function T, and conversely, all but two of the periodic points of T generate ring class fields over suitable imaginary quadratic fields. This gives a dynamical proof of a class number relation originally proved by Deuring. It is conjectured that a similar situation holds for an arbitrary prime p in place of p=2, where the case p=3 has been previously proved by the author, and the case p=5 will be handled in Part II.