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 .)