If 
is a root of the polynomial
equation
 |
where the 
s
are integers and 
satisfies no similar equation of degree 
, then 
is called an algebraic integer of degree 
. An algebraic integer is a special case of an algebraic
number (for which the leading coefficient 
need not equal 1). Radical
integers are a subring of the algebraic integers.
A sum or product of algebraic integers is again an algebraic integer. However, Abel's
impossibility theorem shows that there are algebraic integers of degree 
which are not expressible in terms
of addition, subtraction,
multiplication, division,
and root extraction (the elementary
operations) on rational numbers. In fact,
if elementary operations are allowed on real
numbers only, then there are real numbers which are algebraic integers of degree
3 that cannot be so expressed.
The Gaussian integers are algebraic integers of 
,
since 
are roots of
 |
