Addition law of the tilt $K^\flat$ of a field $K$

Solution 1:

To answer your first question, yes -- this should say "$x_m + y_m\equiv z_m\pmod{p}$ for each $m\geq 0.$" I'm not sure exactly what you mean when you ask "how is one meant to understand the calculation at the bottom of that page?" Is there still something about the calculation that confuses you?

To answer the second question, the limit is meant in the sense of limits of sequences, just like in real analysis. To elaborate, remember that $K$ comes equipped with an absolute value $\left|-\right|_K : K\to \Bbb{R}_{\geq 0}$ which defines a topology on $K,$ with basis given by all open balls $B_\epsilon(x) := \{y\in K\mid |y - x|_K < \epsilon\},$ where $x$ ranges over all of $K$ and $\epsilon$ ranges over all positive reals.

Now, since $x_m, y_m\in K,$ it makes sense to consider the limit of the sequence $\{(x_m + y_m)^{p^m}\}_{m\geq 0}.$ This sequence converges to $\ell\in K$ if, for every $\epsilon > 0,$ there exists $N > 0$ such that $|(x_m + y_m)^{p^m} - \ell|_K < \epsilon$ whenever $m\geq N.$