130 
INTEGRAL ALGEBRAIC NUMBERS [chap, ix 
Theorem i. If d is an integer ^i, not divisible by 
a square > i, all quadratic integers of the field R{\ / d) are 
given by x+yd, where x and y are rational integers and 
6 = V d when d is of one of the forms 4^ + 2, 4Æ+3, while 
d is defined by (2) when d is of the form 4& + 1. 
The quadratic integers of R{V d) are said to have the 
basis 1, d since they are all linear combinations of 1 and d 
with integral coefficients x, y. Note that every number 
of the field is expressible as a linear combination r • 1 + 
sd with rational coefficients r, s. 
Theorem 2. The sum, difference, or product of any 
two quadratic integers of the field R{V d) is a quadratic 
integer. 
For, if x, y, z, w are all integers, the sum of q = x+yd 
and t — z-fwd is r-\-sd, where r = x+z and s = y+ware inte 
gers. Likewise, q — t is a quadratic integer. Finally, the 
product qt is the sum of xz-\-{xw-\-yz)d and ywd 2 , and, 
by the previous result, will be a quadratic integer if 
d 2 , and hence also ywd 2 , is one. The latter is evident 
if 6 — V~d, and is true also for case (2) since then d 2 = d-\-k, 
where k=\{d—i) is an integer. 
82. Algebraic numbers. We shall generalize the 
preceding concepts and theorems. When the coefficients 
of an algebraic equation are all rational numbers, the 
roots are called algebraic numbers. For an equation 
(3) x nJ ra 1 x n ~ 1 -\- . . . . +a n = o 
with integral coefficients, that of the highest power of 
x being unity, the roots are called integral algebraic 
numbers.