CAUTION: You should read the section on permutations
several times if necessary, because it is the basic idea that
makes Galois theory come to life. We have used
basic example. You could play around with
where A has 1,
2, 4, or 100 elements. If A has 100 elements, do not try to
write out all the elements of
. There are 100! of them, and
it would take you almost 3 × 10
years if you wrote out one
element every second.
Digression: Mathematics and Society
Mathematics is like a game. It has rules, and to enjoy playing
or watching it, you have to know and understand the rules.
Mathematicians make up the rules as they go along. There is
sometimes an extrinsic beneﬁt, because mathematics is used for
many practical things, starting with counting and telling time,
and including theoretical physics and constructing computers. Even
number theory has extrinsic beneﬁts nowadays, coming mostly
from the theory of codes and ciphers, but also in acoustics, radar,
and other areas. But even when no applications are immediately
apparent, playing the game can be satisfying.
Many people believe that all of mathematics has already been
discovered and codiﬁed. Mathematicians (they think) do nothing
except rearrange the material in different ways for different types
of students. This seems to be the result of the cut-and-dried method
of teaching mathematics in many high schools and universities.
The facts are laid out in the cleanest logical order. Little attempt
is made to show how someone once had to invent it all, at ﬁrst in
a confused way, and that only later was it possible to give it this
neat form. Many textbooks make no effort to tell about directions
that are still to be explored, conjectures that are unproven, nor, of
course, of ideas that are yet to be formulated.
Project yourself back in time to 1000
BC. Very little mathematics
was known then. It all lay in the future to be discovered, debated,
arranged, and improved. The situation today is nearly the same!