For example an angle of 30 degrees. I know that geometrically I can obtain the entire 30-60-90 triangle using the standard tools (compass, straightedge and unit length) and by performing iterations. However, for this particular problem, I have to use field theoretic machinery. I know that geometric construction problems have analogous problems in field theory involving extensions and irreducible polynomials, I'm just not exactly sure how the two are tied together.
Also, would something similar work for lengths like $\sqrt{2}$?