CHAPTER IX 
INTEGRAL ALGEBRAIC NUMBERS 
80. Purpose of the chapter. We shall develop those 
properties of algebraic numbers which are essential in 
providing an adequate background for the theory of the 
arithmetic of any rational algebra to be presented in the 
next chapter. The latter theory will there be seen 
to be a direct generalization of the theory of algebraic 
numbers. 
In order to make our presentation elementary and 
concrete, we shall develop the theory of quadratic 
numbers before taking up algebraic numbers in general. 
81. Quadratic numbers. Let d be an integer, other 
than +i, which is not divisible by the square of any 
integer >i. As explained in § i, the Jield R(V d) is 
composed of all rational functions of V d with rational 
coefficients. Such a function can evidently be given 
the form 
e-\-f V d 
q ~ g+hVd’ 
where e, /, g, h are rational numbers, and g and h are not 
both zero. Multiplying both numerator and denomina 
tor by g—hVd, in order to rationalize the denominator, 
we obtain q = a-\-bV d, where a and h are rational. Evi 
dently q and a —bl d are the roots of 
(i) x 2 —2ax-\-(a 2 —db 2 )=o, 
whose coefficients are rational. For this reason, q is 
called a quadratic algebraic number. 
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