Algebraic Integer
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
 |
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