Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares?


Example. A positive integer does not have a square root, but is the sum of at most 4 squares. (Lagrange Theorem). However, a real positive number has a square root.


Another Example. A real quadratic form that is postive definite (or semi-definite) is, after a change of coordinates, a sum of squares. How about rational or integral quadratic forms?


Last Example. A positive definite (or semidefinite) real or complex matrix has a square root. How about rational or integral matrices?


Do you have other examples?



When does 'positive' imply 'sum of squares'?
For many examples of this kind, see Olga Taussky, "Sums of squares", *Amer. Math. Monthly* 77 (1970) 805-830.


