Are there countable index subrings of the reals? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:39:53Z http://mathoverflow.net/feeds/question/18877 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18877/are-there-countable-index-subrings-of-the-reals Are there countable index subrings of the reals? Fabrizio Polo 2010-03-20T20:41:09Z 2010-03-21T04:04:47Z <ol> <li><p>Does ${\mathbb R}$ have proper, countable index subrings? By countable I mean finite or countably infinite. By subring I mean any additive subgroup which is closed under multiplication (I don't care if it contains $1$.) By index, I mean index as an additive subgroup.</p></li> <li><p>Given some real number $x$, when is it possible to find a countable index subring of ${\mathbb R}$ which does not contain $x$? </p></li> </ol> http://mathoverflow.net/questions/18877/are-there-countable-index-subrings-of-the-reals/18892#18892 Answer by François G. Dorais for Are there countable index subrings of the reals? François G. Dorais 2010-03-21T01:44:40Z 2010-03-21T03:10:37Z <p><strike>I think the answer is no, but I didn't get anywhere.</strike> [Edit: I used to think the answer was no, but Simon Thomas convinced me otherwise.] Here is the condensed version of what I posted earlier, which seems to put serious constraints on $S$.</p> <p>Let $R$ be any field and let $S$ be an additive subgroup of $R$ which is closed under multiplication. If $S$ has index less than $|S|$ as an additive subgroup of $R$, then every element of $R$ is of the form $a/b$ for $a, b \in S$. To see this, pick $r \in R$ and consider the multiples $ur$ for $u \in S$. Since $S$ has index less than $|S|$, there must be $u \neq v$ such that $a = ur - vr \in S$ then $r = a/b$ where $b = u - v$.</p> http://mathoverflow.net/questions/18877/are-there-countable-index-subrings-of-the-reals/18900#18900 Answer by Simon Thomas for Are there countable index subrings of the reals? Simon Thomas 2010-03-21T02:59:54Z 2010-03-21T02:59:54Z <p>Perhaps surprisingly, it turns out that such subrings do exist. This was proved in Section 2 of my paper:</p> <p>Simon Thomas, Infinite products of finite simple groups II, J. Group Theory 2 (1999), 401--434. </p> <p>The basic idea of the proof is quite simple. Clearly the ring of $p$-adic integers has countable index in the field of $p$-adic numbers. Now the $p$-adic integers are the valuation ring of the obvious valuation on the field of $p$-adic numbers ... and it turns out to be enough to show that $\mathbb{C}$ has an analogous valuation. This is true because $\mathbb{C}$ is isomorphic to the field of Puiseux series over the algebraic closure of $\mathbb{Q}$, which has an appropriate valuation.</p> http://mathoverflow.net/questions/18877/are-there-countable-index-subrings-of-the-reals/18909#18909 Answer by Bjorn Poonen for Are there countable index subrings of the reals? Bjorn Poonen 2010-03-21T04:04:47Z 2010-03-21T04:04:47Z <p>Simon Thomas's approach answers Question 2 too. The answer is that for any nonzero real number $x$ there exists such a subring (possibly without 1) not containing $x$.</p> <p><strong>Proof:</strong> Let $K$ be the Puiseux series field <code>$\overline{\mathbf{Q}}((t^{\mathbf{Q}}))$</code>, let $A$ be its valuation ring, and let $\mathfrak{m}$ be its maximal ideal.</p> <p>If $x \in \mathbf{R}$ is not algebraic over $\mathbf{Q}$, then choose an identification $\mathbf{C} \simeq K$ sending $x$ to the transcendental element $1/t$, and use the subring $\mathbf{R} \cap A$.</p> <p>If $x \in \mathbf{R}^\times$ is algebraic over $\mathbf{Q}$, then choose an identification $\mathbf{C} \simeq K$ again, and use $\mathbf{R} \cap \mathfrak{m}$ (a subring of $\mathbf{R}$ without $1$). $\square$</p> <p><strong>Remark:</strong> If one insists on using subrings with $1$, then the answer is that such a subring not containing $x$ exists if and only if $x \notin \mathbf{Z}$. </p> <p><strong>Proof:</strong> Repeat the argument above, but in the case where $x$ is algebraic (and outside $\mathbf{Z}$), use $\mathbf{R} \cap (\mathbf{Z} + \mathfrak{m})$.</p>