The Scholz reflection principle says, among other things, that if $D < 0$ is a negative fundamental discriminant, not $-3$, then the 3-ranks of the class group of $\mathbb{Q}(\sqrt{D})$ is either equal to that of $\mathbb{Q}(\sqrt{-3D})$, or one larger.


Does anyone know of (and recommend) any introductory reading on this fact? Why it is true, what context to view it in, etc.? Googling reveals some highbrow perspectives on it, some interesting applications, and citations to Scholz's 1932 article (which I'm having difficulty accessing for the moment). All of this is interesting, but there doesn't seem to be any obvious place to begin.


Thank you!



Introductory reading on the Scholz reflection principle?
This is simple class field theory plus Galois theory. Consider a quadratic number field $K$
with class number divisible by $3$. For constructing an unramified cyclic cubic extension $L/K$, adjoin the cube root of unity, and denote the resulting field by $K'$. The Kummer generator of the Kummer extension $L' = K'(\sqrt[3]{\mu})$ must be an ideal cube for the extension to be unramified: $(\mu) = {\mathfrak m}^3$. Since $L'/K$ is abelian, Galois theory shows that the ideal class of ${\mathfrak m}$ must come from the quadratic subfield $F$ different from $K$ and ${\mathbb Q}' = {\mathbb Q}(\sqrt{-3})$. Thus the unramified cubic extensions of $K$ correspond roughly to the $3$-class group of $F$; any differences come from the fact that $\mu$ might be a unit.


The reflection theorem was found independently by Reichardt and then generalized by Leopoldt. 
For a dvi file of Scholz's article, see 
[here](http://www.rzuser.uni-heidelberg.de/~hb3/schbib.html). 


**Edit.**Here's an [English translation](http://www.rzuser.uni-heidelberg.de/~hb3/publ/S11e.pdf).


