The theory of numbers, by Robert D. Carmichael ...

OTHER TOPICS

81

It is found to be possible to introduce in this way a general
set of imaginaries satisfying congruences with prime moduli;
and the new quantities or marks have the property of combining
according to the laws of algebra.
The quantities so introduced are called Galois imaginaries.
We cannot go into a development of the important theory
which is introduced in this way. We shall be content with
indicating two directions in which it leads.
In the first place there is the general Galois field theory
which is of fundamental importance in the study of certain
finite groups. It may be developed from the point of view
indicated here. An excellent exposition, along somewhat
different lines, is to be found in Dickson's Linear Groups with
an Exposition of the Galois Field Theory.
Again, the whole matter may be looked upon from the geometric point of view. In this way we are led to the general
theory of finite geometries, that is, geometries in which there
is only a finite number of points. For a development of the
ideas which arise here see Veblen and Young's Projective
Geometry and the memoir by Veblen and Bussey in the Transactions of the American Mathematical Society, vol. 7, pp.
241-259.
~ 43. ARITHMETIC FORMS
The simplest arithmetic form is ax+b where a and b are
fixed integers different from zero and x is a variable integer.
By varying x in this case we have the terms of an arithmetic
progression. We have already referred to Dirichlet's celebrated theorem which asserts that the form ax+b has an infinite
number of prime values if only a and b are relatively prime.
This is an illustration of one type of theorem connected with
arithmetic forms in general, namely, those in which it is asserted
that numbers of a given form have in addition a given property.
Another type' of theorem is illustrated by a result stated
in ~41, provided that we look at that result in the proper
way. We saw that the number 2 is a quadratic residue of