Also over the rationals (as @KConrad might have mentioned!), the theory of quadratic forms is nicer than the theory of higher forms:

 1. [Hasse’s principle][1] holds for quadratic forms: if a quadratic form has a zero in the reals and in all the p-adics, then it has a rational zero too. By contrast this fails even for ternary cubics, as in [Selmer's example][2].

 2. There are simple questions about length and area which have been open for over 300 years and are equivalent to solubility of quadratic forms over the rationals, with nothing so old and geometric for higher forms. E.g.:

- Is there a [perfect Euler brick][3], whose side lengths and diagonals on the faces and main diagonal all have rational lengths?

- [The congruent number problem][4]: Which numbers can be the area of a right triangle whose sides all have rational lengths?


  [1]: https://en.wikipedia.org/wiki/Hasse_principle#Forms_representing_0
  [2]: https://kconrad.math.uconn.edu/blurbs/gradnumthy/selmerexample.pdf
  [3]: https://en.wikipedia.org/wiki/Euler_brick
  [4]: https://kconrad.math.uconn.edu/articles/congruentnumber.pdf