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
is a root of the polynomial equation 
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.
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. 
, since 
are roots of