My question might be an easy or could be a bit complicate and classic. Actually I am trying to understand why the discriminant of a binary quadratic form is a "the fundamental invariant" under $GL(2,\mathbb{Z})$-action i.e any other invariant is a polynomial of the discriminant. Also I am interested to know about general binary forms. More precisely I would like to know the number of fundamental invariant of binary form of degree n under $GL(2,\mathbb{Z})$-action. 



Invariant theory
Finding the number of generators for the invariants of binary forms is a classic and very hard problem in invariant theory. For forms of small degree one can find a description in Hilbert's book on invariant theory (ISBN 978-0521449038). In the 19th century it was solved for forms of degree at most 8, and it has recently been pushed to degree 10 in <http://dx.doi.org/10.1016/j.jsc.2010.03.002> using computer calculations (where one needs 106 generators). The latter link gives the history of the problem in more detail. 


