Page 292 - DMTH403_ABSTRACT_ALGEBRA

Unit 31: The Galois Group of a Polynomial

For example for n = 17 there is a period that is a sum of eight roots of unity, one that is a sum of Notes
four roots of unity, and one that is the sum of two, which is cos(2/17).
Each of those is a root of a quadratic equation in terms of the one before. Moreover, these
equations have real rather than imaginary roots, so in principle can be solved by geometric
construction: this because the work all goes on inside a totally real field.
In this way the result of Gauss can be understood in current terms; for actual calculation of the
equations to be solved, the periods can be squared and compared with the ‘lower’ periods, in a
quite feasible algorithm.

Compass and Straightedge Constructions

Compass and straightedge constructions are known for all constructible polygons. If n = p· q
with p = 2 or p and q co-prime, an n-gon can be constructed from a p-gon and a q-gon.
If p = 2, draw a q-gon and bisect one of its central angles. From this, a 2q-gon can be

constructed.
If p > 2, inscribe a p-gon and a q-gon in the same circle in such a way that they share a

vertex. Because p and q are relatively prime, there exists integers a,b such that ap + bq = 1.
Then 2a/q + 2b/p = 2/pq. From this, a p·q-gon can be constructed.
Thus one only has to find a compass and straightedge construction for n-gons where n is a
Fermat prime.

The construction for an equilateral triangle is simple and has been known since Antiquity.

Constructions for the regular pentagon were described both by Euclid and by Ptolemy.
Although Gauss proved that the regular 17-gon is constructible, he didn’t actually show

how to do it. The first construction is due to Erchinger, a few years after Gauss’ work.
The first explicit construction of a regular 257-gon was given by Friedrich Julius Richelot

(1832).
A construction for a regular 65537-gon was first given by Johann Gustav Hermes (1894).

The construction is very complex; Hermes spent 10 years completing the 200-page
manuscript. (Conway has cast doubt on the validity of Hermes’ construction, however.

Figure 31.2

LOVELY PROFESSIONAL UNIVERSITY 285