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?