  Solutions of the quartic Fermat equation in ring class fields of odd
conductor over quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ with $-d \equiv 1$
(mod $8$) are shown to be periodic points of a fixed algebraic function $T(z)$
defined on the punctured disk $0< |z|_2 \le \frac{1}{2}$ of the maximal
unramified, algebraic extension $\textsf{K}_2$ of the $2$-adic field
$\mathbb{Q}_2$. 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.
