I'm not clear on exactly what you're asking. If you're willing to accept that each element can be represented as an expression involving field operations and sqrt, then you can build it from the inside out the way you did your example. If you want to make that more formal, you could formalize that notion of expressions and show the numbers represented by are the Pythagorean closure.

Or another approach to formalizing it, you could consider a sequence of fields K_n starting with K_0 = Q where each field contains square roots of all elements in the preview of the sequence. Then prove their union is the Pythagorean closure. Then for each element x you can talk about the smallest n such that x in K_n, which implies x = sqrt(y) for some y in K_(n-1).