Examples
Examples of perfect fields are:
- every field of characteristic zero, e.g. the field of rational numbers or the field of complex numbers;
- every finite field, e.g. the field Fp = Z/pZ where p is a prime number;
- every algebraically closed field;
- the union of perfect fields;
- fields algebraic over a perfect field.
In fact, most fields that appear in practice are perfect. The imperfect case arises mainly in algebraic geometry in characteristic p>0. Every imperfect field is necessarily transcendental over its prime subfield (the minimal subfield), because the latter is perfect. An example of an imperfect field is
- the field of all rational functions in an indeterminate, where k has characteristic p>0 (because X has no p-th root in k(X)).
Read more about this topic: Perfect Field
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