Exercise 1, § 6

Every element z of the p-adic field $\mathbf{Q}_p$ ($p$ a prime number) may be written uniquely as $p^h \sum_{k=0}^{\infty} c_k p^k$, where $h \in \mathbf{Z}$, $c_0 \neq 0$, $0 \leq c_k < p$ for all $k \geq 0$ ("p-adic expansion" of z).

(a) Show that, in order that $z \in \mathbf{Q}$, it is necessary and sufficient that there exist two integers $m \geq 0,\ n \geq 1$ such that $c_{k+n} = c_k$ for all $k \geq m$. (Observe that, if $z = a/b \in \mathbf{Q}$ where $b$ is not divisible by $p$, then

where $a_{n+1}$ is an integer and the sequence $|a_k|$ (ordinary absolute value) is bounded.)

(b) Suppose that the increasing sequence $(k_n)$ of integers $k$ such that $c_k \neq 0$ is such that $\limsup_{n \to m} (k_{n+1}/k_n) = +\infty$. Show that $z$ is transcendental over $\mathbf{Q}$. (For $x \in \mathbf{Q}$, let $|x|$ and $|x|_p$ respectively denote the usual absolute value and a $p$-adic absolute value. Suppose that $P \in \mathbf{Z}[X]$ is an irreducible polynomial such that $P(z) = 0$; if $z_n = p^n \sum_{k=0}^\infty c_k p^k$, show that $|P(z_n)/(z - z_n)|_p$ tends to a limit $\neq 0$ as $n$ tends to $+\infty$, using Taylor's formula; then obtain a contradiction from the hypothesis considering the usual absolute values $|P(z_n)|$.)