Exercise 5, § 3

Let $\omega$ be an absolute value on the field $\mathbf{Q}$ of rational numbers. a) Now prove that $\omega$ is entirely determined by the knowledge of its values for prime numbers. b) If there exists a prime number $p$ such that $\omega(p) \leqslant 1$, now prove that for every other prime number $q$, one has $\omega(q) \leqslant 1$ (majorize $\omega(q^n)$ by writing $q^n$ in the numeral system with base $p$, and make $n$ grow indefinitely). c) If there exists a prime number $p$ such that $\omega(p) < 1$, now prove that for every other prime number $q$, one has $\omega(q) = 1$ (show that one cannot have $\omega(q) < 1$, by using the fact that for every integer $n > 0$, there exist two rational integers $r, s$ such that $1 = rp^n + sq^n$). Deduce that $\omega$ is then an absolute value deduced from the $p$-adic valuation on $\mathbf{Q}$. d) If $\omega(p) > 1$ for every prime number $p$, now prove that if $p$ and $q$ are any two prime numbers, one has

(same method as in b)). Deduce that one then has $\omega(x) = |x|^p$ with $p \leqslant 1$.