Galois Imaginaries

Galois Imaginaries

If one is working in the domain of real numbers the equation x^2 + 1 = 0 has no solution; for there is no real number whose square is -1. If, however, one enlarges the “number system” so as to include not only all real numbers but all complex numbers as well, then it is true that every algebraic equation has a root. It is on account of the existence of this theorem for the enlarged domain that much of the general theory of algebra takes the elegant form in which we know it.

The question naturally arises as to whether we can make a similar extension

in the case of congruences. The congruence x^2 = 3 (mod 5) has no solution, if we employ the term solution in the sense in which we have so far used it. But we may if we choose intro duce an imaginary quantity, or mark, j such that j^2 = 3 (mod 5),

just as in connection with the equation x^2 + 1 = 0 we would introduce the symbol i having the property expressed by the equation i^2 = -1.

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.