Given a field and an extension field, an element is called algebraic over if it is a root of some nonzero polynomial with coefficients in .
Obviously, every element of is algebraic over . Moreover, the sum, difference, product, and quotient of algebraic elements are again algebraic. It follows that the simple extension field is an algebraic extension of iff is algebraic over .
The imaginary unit i is algebraic over the field of real numbers since it is a root of the polynomial . Because its coefficients are integers, it is even true that is algebraic over the field of rational numbers, i.e., it is an algebraic number (and also an algebraic integer). As a consequence, and are algebraic extensions of and respectively. (Here, is the complex field , whereas is the total ring of fractions of the ring of Gaussian integers .)
and an extension field 
, an element 
is called algebraic over 
if it is a root of some nonzero polynomial with coefficients in 
. 
is algebraic over 
is an algebraic extension of 
iff 
is algebraic over 
. 
of real numbers since it is a root of the polynomial 
. Because its coefficients are integers, it is even true that is algebraic over the field 
of rational numbers, i.e., it is an algebraic number (and also an algebraic integer). As a consequence, 
and 
are algebraic extensions of 
and 
respectively. (Here, 
is the complex field 
, whereas 
is the total ring of fractions of the ring of Gaussian integers 
.)