Proposition 2

Let $A$ be a Dedekind domain, $K$ its field of fractions and $P$ the set of non-zero prime ideals of $A$; for $p \in P$ let $v_p$ denote the corresponding essential valuation of $A$. Let $p_1, \ldots, p_q$ be distinct elements of $P$ and $n_1, \ldots, n_q$ rational integers and $x_1, \ldots, x_q$ elements of $K$. Then there exists $x \in K$ such that $v_{p_i}(x - x_i) \geq n_i$ for $1 \leq i \leq q$ and $v_p(x) \geq 0$ for all $p \in P$ distinct from the $p_i$.

Replacing if need be the $n_i$ by greater integers, they may be assumed all to be positive. We examine first the case where the $x_i$ are in $A$; it obviously amounts to finding an $x \in A$ satisfying the congruences

and the existence of $x$ then follows from Chapter II, § 1, no. 2, Proposition 5.

We pass now to the general case. We may write $x_i = s^{-1} y_i$ where $s, y_i$ are in $A$; writing $x = s^{-1} y$, it amounts to finding a $y \in A$ such that, on the one hand, $v_{p_i}(y - y_i) \geq n_i + v_{p_i}(s)$ and, on the other, $v_p(y) \geq v_p(s)$ for all $p \in P$ distinct from the $p_i$; as $v_p(s) = 0$ except for a finite number of indices $p$, it is thus reduced to the above case; whence the proposition.

Proposition 2 may be interpreted as a density theorem. To be precise, for all $p \in P$, let $\hat{K}_p$ (resp. $\hat{A}_p$) be the completion of $K$ (resp. $A$) with respect to the discrete valuation $v_p$ and consider the product $\prod_{p \in P} \hat{K}_p$; an element $x = (x_p)$ of this product is called a restricted adèle of $A$ if $x_p \in \hat{A}_p$ for all $p \in P$ with the exception of a finite number of them. Clearly the set $A$ of restricted adtles is a subring of $\prod_{p \in P} \hat{K}_p$, which contains the product ring $A, = \prod_{p \in P} \hat{A}_p$. Consider on $A$, the product topology, with respect to which $A,$ is complete; there is on $A$ a unique topology $\mathscr{T}$ which is compatible with its additive group structure and for which the neighbourhoods of 0 in $A$, form a fundamental system $\mathscr{G}$ of neighbourhoods of 0. The topology $\mathscr{T}$ is compatible with the ring structure on $A$; for clearly axiom (AVII) of General Topology, Chapter 111, § 6, no. 3, holds, the topology induced by $\mathscr{T}$ on $A,$ being compatible with the ring structure on $A,$. On the other hand, for all $x \in A$ there exists a finite subset $J$ of $P$ such that, if we write $J' = P - J, \ K_J \leq \prod_{p \in J} \hat{K}_p, \ A_{J'} = \prod_{p \in J'} \hat{A}_p,$ then x \in K_J \times A_{J'} and, as $\hat{A}_p$ is open in $\hat{K}_p$ for all p, $\mathscr{G}$ is a fundamental system of neighbourhoods of 0 for the product topology on $K_J \times A_{J'}$; since the latter is compatible with the ring structure on this product, axiom (AV_1) of General Topology, Chapter III, loc. cit. is also seen to hold, which proves our assertion. Clearly A, is an open subring of A and hence A is also a complete ring (General Topology, Chapter III, § 3, no. 3, Proposition 4).

For all $x \in K$, let $\Delta(x)$ be the element $(x_p) \in \prod_{p \in P} \hat{K}_p$ such that $x_p = x$ for all $p \in P$; as $x_p \in \hat{A}_p$ except for a finite number of values of p, $\Delta(x) \in A$; hence we have thus defined a homomorphism $A : K \to A$ which is injective if $P \neq \varnothing$ (that is if A is not a field); the elements of $\Delta(K)$ are called principal restricted adèles and clearly $\Delta(A) \subset A$.