Ordered basis of a complete p-valued group
= Ordered basis of a complete p-valued group
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An ordered basis $(g_1,\ldots,g_d)$ of a complete finite-rank p-valued group gives each element a unique convergent expression $g_1^{\lambda_1}\cdots g_d^{\lambda_d}$ with $\lambda_i\in\mathbb Z_p$, and its valuation is the minimum of $\omega(g_i)+v_p(\lambda_i)$ over the nonzero coordinates.