Exercise 1, § 8

(a) An element $\pi$ of $K$ is called admissible if $v(\pi) = 0$ and $\pi^{p-1}/p \equiv -1 \pmod{\pi \mathfrak{o}_K}$. (Note that $\pi^{p-1}/p \in \mathfrak{o}_K$ since $v(\pi) = 0$.) Show that there exist fields $K$ satisfying the conditions of the paragraph and containing an admissible element (adjoin to $\mathbf{Q}_p$ a $(p-1)$-th root of $-p$).

(b) Let $\pi$ be an admissible element of $K$. Consider the following formal power series with coefficients in $K$:

Show that these series are inverses of one another and that their coefficients belong to $\mathfrak{o}_K$ (use Lemma 2).

(c) Let $\{U, V\}$ be a set with two elements and let $H(U, V)$ be the Hausdorff series. We write

Then $H_{\pi}(U, V) \in \hat{L}_K(\{U, V\})$.

Show that

Deduce that $H_{\pi} \in \hat{L}_{\mathfrak{o}_K}(\{U, V\})$, i.e. that the coefficients of $H_{\pi}$ belong to $\mathfrak{o}_K$; hence another proof of Proposition 1.

(d) Let $\tilde{e}_{\pi}$, $\tilde{l}_{\pi}$ and $\tilde{H}_{\pi}$ denote the series obtained by reducing $e_{\pi}$, $l_{\pi}$ and $H_{\pi}$ modulo $\pi \mathfrak{o}_K$; their coefficients belong to the ring $\mathfrak{o}_K/\pi \mathfrak{o}_K$.

Show that $\tilde{l}_{\pi}(U) = U - U^p$ (note that $v_p(n) < (n-1)\theta$ if $n \neq 1,\ p$). Deduce that

and that

or also

where $\Lambda_p$ is defined by the formula

In particular, $\tilde{H}_{\pi}(U, V)$ belongs to $\hat{L}_{\mathbf{F}_p}(\{U, V\})$ and its homogeneous component of degree $p$ is $-\Lambda_p(U, V)$. Then

where $W$ is a third indeterminate.

(e) Let $C(U, V) = H(-U, H(-V, H(U, V)))$ ("Hausdorff commutator"). Then $\exp(C(U, V)) = \exp(-U) \exp(-V) \exp(U) \exp(V)$. We write

Show that the coefficients of $C_{\pi}(U, V)$ belong to $\pi \mathfrak{o}_K$ (use the fact that $\tilde{H}_{\pi}(U, V) = \tilde{H}_{\pi}(V, U)$).†