Page 290 - DMTH403_ABSTRACT_ALGEBRA

Unit 31: The Galois Group of a Polynomial

(Littlewood 1958, Cadogan 1971). In particular, Notes

n
S (x ,..., x ) =  x   1 ...(27)
k
n
1
1
k 1

2
S (x ,..., x ) =   2 2 ...(28)
1
1
n
2
3
S (x ,..., x ) =   3   3 3 ...(29)
1
3
2
1
n
1
2
2
4
S (x ,..., x ) =   4   2  4   4 4 ...(30)
1
n
3
1
1
2
4
1
2
(Schroeppel 1972), as can be verified by plugging in and multiplying through.
31.4 Constructible Polygon
In mathematics, a constructible polygon is a regular polygon that can be constructed with
compass and straightedge. For example, a regular pentagon is constructible with compass and
straightedge while a regular heptagon is not.
Conditions for Constructibility
Some regular polygons are easy to construct with compass and straightedge; others are not. This
led to the question being posed: is it possible to construct all regular n-gons with compass and
straightedge? If not, which n-gons are constructible and which are not?
Carl Friedrich Gauss proved the constructability of the regular 17-gon in 1796. Five years later,
he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae. This theory
allowed him to formulate a sufficient condition for the constructability of regular polygons.
A regular n-gon can be constructed with compass and straight edge if n is the product of a power
of 2 and any number of distinct Fermat primes.
Gauss stated without proof that this condition was also necessary, but never published his proof.
A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as the Gauss–
Wantzel theorem.
Figure 3.1

LOVELY PROFESSIONAL UNIVERSITY 283