Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.2

Theorem

Let $\norm {\, \cdot \,}$ be a non-trivial non-Archimedean norm on the rational numbers $\Q$.

Let $n_0 = \min \set {n \in N : \norm n < 1}$.

Then:

$n_0$ is a prime number.

Proof

Aiming for a contradiction, suppose $n_0$ is a composite number.

Let $n_1, n_2 \in \N$ such that $n_1, n_2 < n_0$ and $n_0 = n_1 n_2$.

By the definition of $n_0$ then:

$\norm {n_1} = 1$

$\norm {n_2} = 1$

By Non-Archimedean Norm Axiom $\text N 2$: Multiplicativity:

$\norm {n_0} = \norm {n_1 n_2} = \norm {n_1} \norm {n_2} = 1$

This contradicts the assumption that $\norm {n_0} < 1$.

So $n_0$ must be a prime number.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.1$ Absolute Values on $\Q$, Theorem $3.1.3$
• 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.9$ Metrics and norms on the rational numbers. Ostrowski’s Theorem, Theorem $1.50$