EAN: 9781461272854

INTRODUCTION TO THE GALOIS CORRESPONDENCE - MAU...

In this presentation of the Galois correspondence  modern theories of groups and fields are
used to study problems  some of which date back to the ancient Greeks. The techniques used to
solve these problems  rather than the solutions themselves  are of primary importance. The
ancient Greeks were concerned with constructibility problems. For example  they tried to
determine if it was possible  using straightedge and compass alone  to perform any of the
following tasks? (1) Double an arbitrary cube  in particular  construct a cube with volume
twice that of the unit cube. (2) Trisect an arbitrary angle. (3) Square an arbitrary circle  in
particular  construct a square with area 1r. (4) Construct a regular polygon with n sides for n
2. If we define a real number c to be constructible if  and only if  the point (c  0) can be
constructed starting with the points (0 0) and (1 0)  then we may show that the set of
constructible numbers is a subfield of the field R of real numbers containing the field Q of
rational numbers. Such a subfield is called an intermediate field of Rover Q. We may thus gain
insight into the constructibility problems by studying intermediate fields of Rover Q. In
chapter 4 we will show that (1) through (3) are not possible and we will determine necessary
and sufficient conditions that the integer n must satisfy in order that a regular polygon with
n sides be constructible.

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