# Binary quadratic forms over Z and class numbers of quadratic ﬁelds.

What is the relation between the classiﬁcation of binary quadratic forms over $\mathbb Z$, and the problem of ﬁnding the class numbers of quadratic ﬁelds?

What would be a nice reference for this?

For a very beautiful, modern, and general treatment, see this paper of Melanie Matchett Wood. The classical comparison is explained at the beginning of the paper, and then it goes on to develop very general results. I think it is a very beautiful treatment of the subject; even the classical statements are made much clearer than is usual (in my experience) in more traditional treatments.

• Thanks for sharing that link. It is indeed a very nice exposition. There are some older treatments that go partway there, but I cannot recall them off the top of my head. Oct 18, 2012 at 14:37

My favourite reference on this subject would be the excellent treatment contained in the book "Primes of the form $x^2+ny^2$ by D. A. Cox - see here

He gives a lot of details of the historical background (going back to Fermat and Euler) to both binary quadratic forms and the class number problem for quadratic fields. It is not really a textbook, but is very readable with many interesting exercises, and a huge collection of references for further study.

• Thank you very much! I will try to get hold of this book Oct 8, 2012 at 22:34
• For a more modern treatment (but also much much more sophisticated), you might look at J-P, Serre's book "A Course in Arithmetic" - see here, but this book requires much more by way of algebraic knowledge. Oct 8, 2012 at 22:38

Along with Cox, I like Duncan Buell, Binary Quadratic Forms because it does pretty much everything with integral forms, including indefinite forms and those with odd $b$ in $f(x,y) = a x^2 + b x y + c y^2.$ It gives explicit calculations for composition, particularly computer algorithms. see HERE

• Notice that computations for the composition go back to Gauss, or earlier. Feb 2, 2013 at 4:53
• I have been wondering whether explaining (in possibly somewhat informal style) the connection between QFs and class numbers would make a fine topic for a blog post. In that case you would be the logical choice to author it. I guess many members like me have only seen the class group way of thinking about it. The references are there, sure, but still? Jun 17, 2014 at 9:33