Also over the rationals (as @KConrad might have mentioned!), the theory of quadratic forms is nicer than the theory of higher forms:
Hasse’s principle 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. (See the Wikipedia article and Conrad - Selmer's example.)
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, whose side lengths and diagonals on the faces and main diagonal all have rational lengths?
Which numbers are congruent (see Conrad - The congruent number problem), in the sense of being the area of a right triangle whose sides all have rational lengths?
