An introduction to the modern theory of equations, by Florian Cajori.

CHAPTER XIII *

THE GALOIS THEORY OF ALGEBRAIC NUMBERS. REDUCIBILITY
120. Definition of Domain. A set of numbers is called a
domain of rationality or simply a domain, when the sums, differences, products, and quotients of any numbers in the set
(excluding only the quotients obtained through division by 0)
always yield as results numbers belonging to the set.
All rational numbers (integers and rational fractions, taken
both positively and negatively) constitute such a domain, for
this system of magnitudes is complete in itself in the sense
that any of the four operations involving any of these numbers
never yields as a result a number which does not belong to
the set.
The integers by themselves do not constitute a domain, for
the quotient of two integers may be fractional.
All the numbers of one domain may be contained in a
second and larger domain 2'. In this event the smaller domain
0 is called a divisor of the other Q', and i2' is called a domain
over 0.
For example, the complex numbers of the form a + ib, where
i =/- 1 and a and b signify rational numbers, are a domain
of which the domain of rational numbers is a divisor.
Another example of domains of numbers is the one embracing all real numbers, whether rational or irrational. Still
another is the domain consisting of all numbers, a + ib, where
a and b are rational or irrational.
* In the exposition of the Galois theory in this and the succeeding chapters
we have followed the treatment given by H. Weber in his Lehrbu.ch der
Algebra, Vol. I, pp. 491-698.
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