User lyx - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:37:35Z http://mathoverflow.net/feeds/user/19463 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81626/is-strong-approximation-difficult/81676#81676 Answer by LYX for Is strong approximation difficult? LYX 2011-11-23T01:35:04Z 2011-11-23T01:35:04Z <p>I am quite surprised that it seems to be so poorly known, but an elementary proof of Strong Approximation for $SL_n$ over a Dedekind domain can already be found in Bourbaki (Algebre Commutative, VII, $\S$2, n.4). Essentially, the idea is to deduce it from the Chinese Remainder Theorem, by means of elementary matrices, as hinted in Ralph's answer. Actually, I would be curious to know a bit more about the history of this case of Strong Approximation: who and when did prove it first?</p> http://mathoverflow.net/questions/81626/is-strong-approximation-difficult/81676#81676 Comment by LYX LYX 2011-11-23T02:42:12Z 2011-11-23T02:42:12Z Bourbaki's Algebre Commutative, VII, &#167;2, n.4 has the title: &quot;The Approximation Theorem for Dedekind domains&quot;. They take a Dedekind domain $A$ with fraction field $K$ and define the ring of restricted adeles $\bf A$ as the restricted product of completions of $K$ over all (non-zero) primes of $A$. Proposition 4 states: $SL(n,K)$ is dense in $SL(n,{\bf A})$.