Use the fact that satisfies the polynomial (to be proved later) to argue that the regular 5-gon is constructible using a straightedge and compass.
Using the rational root test, we can see that is irreducible over . Thus the roots of lie in a degree 2 extension of , and we have seen that all such numbers are constructible by straightedge and compass. So , and hence , is constructible.
Recall that , and so (since is in the first quadrant) . In particular, is also constructible.
I’ll try to describe the rest in words, because my geometric diagrams tend to look like garbage unless I spend a couple of hours on them.
Suppose now that we have a line segment , which we want to be an edge of a regular 5-gon. Extend the line and construct the line perpendicular to at . On , construct the point such that and has measure . On the perpendicular, construct the point such that has measure . Now construct the perpendicular to at and to at , and let be the intersection of these two lines. Finally, construct the point on such that either or and such that and have the same measure. By construction, has measure , so that has measure . Repeat this construction with (and so on), being careful with the orientation, to construct a regular 5-gon.