Lemma 2

$v(n) \leqslant v(n!) \leqslant (n - 1)\theta$ and $v(n) \leqslant (\log n)/(\log p)$ for every integer $n \geqslant 1$. $v(n!) = v_p(n!) = (n - S(n))\theta \leqslant (n - 1)\theta$ by Lemma 1. On the other hand, $n \geqslant p^{v(n)}$, whence $v(n) \leqslant (\log n)/(\log p)$. Let $I = \{ U, V \}$ be a set of two elements and let

be the Hausdorff series (§ 6, no. 4, Definition 1). Let $\mathbf{Z}_{(p)}$ be the local ring of $\mathbf{Z}$ relative to the prime ideal $(p)$ and $(e_b)_{b \in B}$ a basis of $\mathbf{L}_{\mathbf{Z}_{(p)}}(I)$ over $\mathbf{Z}$ (§ 2, no. 11, Theorem 1). It is also a basis of $\mathbf{L}_{\mathbf{Q}}(I)$ over $\mathbf{Q}$.