If this is minimization subject to some constraints (for example various sums of variables being given), then it is probably accessible via the method of Lagrange multipliers. As it is, the question doesn't seem properly posed.

Please edit this display of irrelevant considerations.

Quite easy to see that totally real fields are so generated. It is also easy to see that the field generated can be strictly larger than a maximal totally real subfield.

Not, I think, a view explicit in the question; but of course the only way to find another John Conway would be cloning technology. High-degree polynomials do occur outside "contrived settings". For some problems such as class field towers (Golod-Shafarevich) they are there, and the question would be why anyone would write them down.

My argument also - but cos not sin?

Roth's result extended the work of Thue and Siegel in diophantine approximation. It was Baker's work that extended what Gelfond did (from two logarithms of algebraic numbers not commensurable to lower bounds for any linear form in logarithms). I think the point here is that auxiliary functions in several complex variables were needed, and new ideas.