I am able to construct the points that trisect a line with ruler and compass. But when trying to give some upper bound degree of the field extensions corresponding to the points trisecting a line I get lost.
The way I constructed this "trisection" was by drawing some arbitrary line segment $p_1p_2$ drawing some other line segment $p_1p_3$. Construct the line segment $p_1p_4$ of length 3$p_1p_3$ and using that "common" parallel technique to trisect the line segment $p_1p_2$.
So, when constructing all the circles, line segments, must I keep track of each segment length to determine the upper bound for degree? Or is there some other simple method that doesn't require such an exhaustive approach?