Probability of an extension being normal - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T06:43:58Z http://mathoverflow.net/feeds/question/28587 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28587/probability-of-an-extension-being-normal Probability of an extension being normal Daniel Barter 2010-06-18T00:35:26Z 2010-06-29T01:46:02Z <p>Let \$P\$ be the probability that an elliptic curve with a rational point has an infinite number of rational points. From what I understand, the value of P is unknown. </p> <p>This got me thinking about a similar question related to Galois theory. Let \$N\$ be the probability that a finite extension of \$\mathbb{Q}\$ is a normal extension. Is anything known about the value of \$N\$?. </p> <p>I wouldn't expect it to be 1 or 0. </p> <p>My gut feeling is that not much is known as it seems related to the inverse Galois problem. Indeed, if every finite group were the galois group of some finite galois extension of \$\mathbb{Q}\$ i would imagine that \$R\$ would be somehow related to the "probability of a subgroup being normal" (this is intentionally vague).</p> <p>Question: Is anything (if anything) known about \$N\$?</p> http://mathoverflow.net/questions/28587/probability-of-an-extension-being-normal/29854#29854 Answer by JSE for Probability of an extension being normal JSE 2010-06-29T01:46:02Z 2010-06-29T01:46:02Z <p>The probability is 0, if you count number fields of some fixed degree in order of discriminant. Venkatesh and I prove in <a href="http://arxiv.org/abs/math.NT/0309153" rel="nofollow">this paper</a> (later published in Ann. of Math., 163, no.2, 2006) that there are at most c_n,e X^{3/8+e} Galois extensions of degree n and discriminant at most X, while there are at least c_n X^{1/2} degree-n extensions of discriminant at most X in all. (Theorem 1.1 and Prop 1.3).</p>