A subfield which is strictly smaller than the field in which it is contained.
The field of rationals 
is a proper subfield of the field of real numbers 
which, in turn, is a proper subfield of 
; 
is actually the biggest proper subfield of 
, whereas there are infinite sequences of proper subfields
between 
and 
.
Here is one example, constructed by using the 
th root of 2 for different prime numbers 
,
![Q subset Q[sqrt(2)] subset Q[sqrt(2),RadicalBox[2, 3]] subset Q[sqrt(2),RadicalBox[2, 3],RadicalBox[2, 5],] subset Q[sqrt(2),RadicalBox[2, 3],RadicalBox[2, 5],RadicalBox[2, 7]] subset
Q[sqrt(2),RadicalBox[2, 3],RadicalBox[2, 5],RadicalBox[2, 7],RadicalBox[2, 11]] subset ... subset R.](/images/equations/ProperSubfield/NumberedEquation1.svg) |
Note that all the fields in the sequence are contained in the set of algebraic numbers, which is another proper subfield of 
.
Hence, 
has infinitely many proper subfields. On the contrary, 
has none, since any subfield of 
must contain 0, all integer multiples of 1, and all their
quotients (since every field is a division algebra),
thus generating all the rational numbers. In particular, 
is the smallest proper subfield of 
.
For all prime numbers 
and integers 
,
the prime field 
is a proper subfield of 
.
