A subfield which is strictly smaller than the field in which it is contained. The field of rationals Q is a proper subfield of the field of real numbers R which, in turn, is a proper subfield of C; R is actually the biggest proper subfield of C, whereas there are infinite sequences of proper subfields between Q and R. Here is one example, constructed by using the pth root of 2 for different prime numbers p, Q subset Q[sqrt(2)] subset Q[sqrt(2), 2^(1/3)] subset Q[sqrt(2), 2^(1/3), 2^(1/5), ] subset Q[sqrt(2), 2^(1/3), 2^(1/5), 2^(1/7)] subset Q[sqrt(2), 2^(1/3), 2^(1/5), 2^(1/7), 2^(1/11)] subset ... subset R. Note that all the fields in the sequence are contained in the set of algebraic numbers, which is another proper subfield of R.

field | prime subfield | subfield