Proposition 7

Let us retain the hypotheses and notations of th. 2. There exists a unique multiplicative subset S of A such that $\pi$ induces a bijection of S onto k. In order that an element a of A belong to S, it is necessary and sufficient that for every $n \in \mathbf{N}$, there exist an element $a_n$ of A such that $a = a_n^{p^n}$. The set S is the set of elements of the form $u(x, 0, 0, ...)$.

Let us first prove the uniqueness of S. Let S be a multiplicative subset of A, such that $\pi$ induces a bijection of S onto k. Let T be the set of elements of A which are $p^n$-th powers for every $n \in \mathbf{N}$.

a) One has $S \subset T$ : Let $a \in S$ and $n \in \mathbf{N}$; since the field k is perfect, there exists an element $x_n$ of k such that $x_n^{p^n} = \pi(a)$; since one has $\pi(S) = k$, there exists an element $a_n$ of S such that $x_n = \pi(a_n)$. One then has $\pi(a_n^{p^n}) = \pi(a)$ whence $a_n^{p^n} = a$ since the restriction of $\pi$ to S is injective.

b) The restriction of $\pi$ to T is injective: let a and b be two elements of T such that $\pi(a) = \pi(b)$. Let $n \in \mathbf{N}$; there exist two elements $a_n$ and $b_n$ of A such that $a = a_n^{p^n}, b = b_n^{p^n}$. We then have $\pi(a_n)^{p^n} = \pi(b_n)^{p^n}$, whence $\pi(a_n) = \pi(b_n)$, that is to say $a_n \equiv b_n$ mod. $m_A$. By Lemma 1 of § 1, No. 1, we have $a_n^{p^n} \equiv b_n^{p^n}$ mod. $m_A^{n+1}$ that is to say $a \equiv b$ mod. $m_A^{n+1}$. Since n is arbitrary, we have $a = b$.

Properties a) and b) above, together with the formula $\pi(S) = k$, imply the relation $S = T$, whence uniqueness.

Let us now prove the existence of S. With the notation of Theorem 2, put $\varphi = u \circ \tau_k$, that is to say (§ 1, No. 6)

(2)

for every $x \in k$. By Proposition 4 of loc. cit., we have

(3)

It is clear that the mapping $\pi \circ \varphi$ is the identity mapping of $k$. Therefore the image S of $\varphi$ satisfies the conditions of Proposition 7.

The elements of S are often called the multiplicative (or Teichmüller) representatives of A.