Page 291 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Construction of the regular 257-gon
Detailed results by Gauss’ theory

Only five Fermat primes are known:
F = 3, F = 5, F = 17, F = 257, and F = 65537 (sequence A019434 in OEIS)
1
2
3
0
4
The next twenty-eight Fermat numbers, F through F , are known to be composite.
5
32
Thus an n-gon is constructible if
n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, (sequence A003401 in OEIS),
while an n-gon is not constructible with compass and straightedge if
n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, (sequence A004169 in OEIS).
31.5 Connection to Pascal’s Triangle

There are 31 known numbers that are multiples of distinct Fermat primes, which correspond to
the 31 odd-sided regular polygons that are known to be constructible. These are 3, 5, 15, 17, 51,
85, 255, 257, , 4294967295. As John Conway commented in The Book of Numbers, these numbers,
when written in binary, are equal to the first 32 rows of the modulo-2 Pascal’s triangle, minus
the top row. This pattern breaks down after there, as the 6th Fermat number is composite, so the
following rows do not correspond to constructible polygons. It is unknown whether any more
Fermat primes exist, and is therefore unknown how many odd-sided constructible polygons
exist. In general, if there are x Fermat primes, then there are 2 ”1 odd-sided constructible polygons.
x
General Theory

In the light of later work on Galois Theory, the principles of these proofs have been clarified.
It is straightforward to show from analytic geometry that constructible lengths must come from
base lengths by the solution of some sequence of quadratic equations. In terms of field theory,
such lengths must be contained in a field extension generated by a tower of quadratic extensions.
It follows that a field generated by constructions will always have degree over the base field that
is a power of two.

In the specific case of a regular n-gon, the question reduces to the question of constructing a
length
cos(2/n).

This number lies in the n-th cyclotomic field — and in fact in its real subfield, which is a totally
real field and a rational vector space of dimension
½(n),

where (n) is Euler’s quotient function. Wantzel’s result comes down to a calculation showing
that (n) is a power of 2 precisely in the cases specified.
As for the construction of Gauss, when the Galois group is 2-group it follows that it has a
sequence of subgroups of orders
1, 2, 4, 8, ...
that are nested, each in the next something simple to prove by induction in this case of an abelian
group. Therefore, there are subfields nested inside the cyclotomic field, each of degree 2 over the
one before. Generators for each such field can be written down by Gaussian period theory.

284 LOVELY PROFESSIONAL UNIVERSITY