What are Mean Values of Ideal Densities in Galois Extensions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:56:22Z http://mathoverflow.net/feeds/question/18932 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18932/what-are-mean-values-of-ideal-densities-in-galois-extensions What are Mean Values of Ideal Densities in Galois Extensions? Franz Lemmermeyer 2010-03-21T15:50:23Z 2010-07-22T14:00:55Z <p>In an unfinished (and as of now unpublished) article intended for the encyclopedia of mathematics, Arnold Scholz wrote:</p> <p>"Classifying extensions according to the Galois group of their normal closure provides us with a new point of view. Not only the minimal discriminants but also the mean values of the ideal densities differ considerably, and have the following values for discriminants with large prime factors: </p> <ul> <li>$\sqrt{\zeta(2)}$ for quadratic extensions;</li> <li>$\sqrt[3]{\zeta(3)^2}$ and $\sqrt{\zeta(2)}\sqrt[3]{\zeta(3)}$ for a cubic extension according as it is cyclic or noncyclic;</li> <li>$\sqrt{\zeta(2)}\sqrt{\zeta(4)}$ and $\sqrt{\zeta(2)}^3$ for cyclic and biquadratic quartic extensions, respectively."</li> </ul> <p>I'd like to know what Scholz is talking about here. Ideal density might be some limit of the form "number of ideals with norm $\le x$" / $x$, and mean value should denote some average over number fields. But what exactly is Scholz doing here? </p> <p><b>Edit.</b> Apparently (this is suggested by some remarks he made elsewhere), Scholz called the expression $$\prod_p \Phi_K(p)/\phi(p)$$ the ideal density of a number field $K$, where $\phi$ and $\Phi_K$ denote Euler's phi function in the rationals and in $K$, respectively. This expression occurs in the product formula for the zeta function. I still don't know where to go from here.</p> <p>As for Robin's remark on the density of fields ordered by discriminants, Scholz claimed, in a letter to Hasse dated Sept. 27, 1938, the following: The Dirichlet series $$G(s) = \sum_{Gal(K)=G} D_K^{-s},$$ where the sum is over all quartic fields whose normal closure has Galois group $G$, have abscissa of convergence $\alpha(D)=1$, $\alpha(Z) = \alpha(V) = \frac{1}{2}$ and probably $\alpha(S)=1$, $\alpha(A)=\frac{1}{2}$, where $D$, $Z$, $V$, $A$, $S$ denote the dihedral, cyclic, four, alternating and symmetric group. Moreover, $$\lim_{s \to 1/2} \frac{Z(s)}{V(s)} = 0,$$ where $Z(s)$ and $V(s)$ are the Dirichlet series defined above for $G=Z$ and $G=V$. This is all correct, as we know now, but how could Scholz have discovered (and, for $G = D$, $Z$, $V$, proved) these results in the 1930s? </p> http://mathoverflow.net/questions/18932/what-are-mean-values-of-ideal-densities-in-galois-extensions/18936#18936 Answer by Robin Chapman for What are Mean Values of Ideal Densities in Galois Extensions? Robin Chapman 2010-03-21T16:35:18Z 2010-03-21T16:35:18Z <p>This may be a reference to the the Davenport-Heilbronn theorem on the distribution of discriminants of number fields. See math.stanford.edu/~fthorne/davenport-heilbronn.pdf for a nice exposition. Strictly speaking the Davenport-Heilbronn theorem is the case of cubic extensions. Quadratic extensions are easy. I believe general quartic (and quintic) extensions were proved recently by Bhargava; I don't know who first proved the special cases of quartics that Scholz lists.</p>