Prove Ostrowski's theorem for \(\mathbb{F}_{q}(t)\): every nontrivial absolute value on \(\mathbb{F}_{q}(t)\) is equivalent to \(|\cdot|_{\infty}\) or \(|\cdot|_{\pi}\) for some prime \(\pi\in\mathbb{F}_{q}[t]\). More precisely, show that if \(\|\cdot\|\) is a nontrivial absolute value on \(\mathbb{F}_{q}(t)\), either \(\|t\|>1\) and \(\|\cdot\|\sim|\cdot|_{\infty}\), or \(\|t\|\leq 1\) and \(\|\cdot\|\sim|\cdot|_{\pi}\) for some prime \(\pi\in\mathbb{F}_{q}[t]\).