Exercise 2, § 6

Let $\Gamma$ be a finite commutative monoid. Consider the following conditions $E_0$ and $E_p$ where $p$ denotes a prime number. $(E_0)$ If $x \in \Gamma$ and if $e, d$ are integers $\geq 0$, the relation $x^{e+d} = x^d$ implies the relation $x^{e+1} = x$. $(E_p)$ If $x, y \in \Gamma$, the relation $x^p = y^p$ implies $x = y$. Let $K$ be a field of characteristic $p$ ($\geq 0$) and let $A$ be the $K$-algebra $K^{(\Gamma)}$ of the monoid $\Gamma$. a) For $A$ to be etale it is necessary and sufficient that the condition $(E_p)$ be satisfied. (Use Ex. 1 for the sufficiency. For the necessity observe for $p = 0$ that $x^{e+1} - x$ is nilpotent if $x^{e+d} = x^d$; for $p > 0$ use the Cor. of V, p. 35.) b) Let $X$ be the set of homomorphisms of $\Gamma$ into the multiplicative monoid of $K$. We have $\mathrm{Card}\ X \leq \mathrm{Card}\ \Gamma$. If $K$ is algebraically closed, then $\mathrm{Card}\ X = \mathrm{Card}\ \Gamma$ holds if and only if condition $(E_p)$ is satisfied. c) If $\Gamma$ is a group, condition $(E_0)$ always holds ; if $p > 0$, condition $(E_p)$ is equivalent to saying that each element of $\Gamma$ has order prime to $p$, or also that $\mathrm{Card}\ \Gamma$ is not divisible by $p$.