Gauss’ Strategy for Constructing the 17-gon

22 Apr

In the previous post, we described how to compute the Galois group of the splitting field of a cyclotomic polynomial, and use it to characterize the subfields of the splitting field.

Gauss used this theorem to solve the ancient problem of constructing the regular 17-gon with compass and straightedge. The Greeks had been able to construct equilateral triangle, the square, and the regular pentagon with compass and straightedge, but didn’t know how to go further. In fact, it’s not possible to construct the 7-gon, the 11-gon, or the 13-gon with straightedge and compass, but it is possible to construct the 17-gon, as Gauss did.

To this end, consider the regular 17-gon with vertices on the unit circle, and with one vertex at (1, 0). The vertices cut the unit circle into 17 arcs of equal length.

If one can construct a segment of length

\large \cos(\frac{2 \pi}{17})

then one can place it along the x-axis of the coordinate plane starting at the origin. If one draws a vertical perpendicular, the point of intersection of this perpendicular with the circle will be the next vertex of the 17-gon (if one orients the vertices counterclockwise), which one can then use to construct the entire regular 17-gon.  

The tools of straightedge and compass allow one to add, subtract, multiply, divide, and take square roots of lengths of segments. Thus, the problem of constructing the 17-gon is the same as that of expressing

\large \cos(\frac{2 \pi}{17})

by successively performing such operations.

 The connection with the 17-th cyclotomic polynomial is that the roots of this polynomial are positioned at the vertices of the 17-gon. This follows from the fact that multiplying two complex numbers on the unit circle yields the point on the unit circle obtained by rotating (1, 0) through the sum of the angles that the points make with the positive x-axis. The fact implies that the roots of the 17-th cyclotomic polynomial are the points [other than (1,0)] that end up (1, 0) when rotated 17 times, and these are the vertices of the 17-gon.

Let ‘z’ the complex root of the 17-th cyclotomic polynomial that’s immediately counterclockwise of (1, 0) on the unit circle. Then the 16-th power of z is the complex root of the 17-th cyclotomic polynomial immediately clockwise of (1, 0), and we have

\large z + z^{16} = \cos(\frac{2 \pi}{17})

 So it suffices to express the left-hand side by successively performing the operations mentioned two paragraphs above.

Galois theory provides a systematic framework for thinking about using the operations mentioned above to break the symmetry present in the coefficients of the 17-th cyclotomic polynomial, in order to find expressions for its roots. We’ll get into the technical details next time.

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