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Let F be a field. A subset k of F is a subfield of F if it is a subring of F and is itself a field. Equivalently a subset k of F is a subfield if and only if
  • 1F in k
  • a-b in k, for all a,b in k
  • ab in k, for all a,b in k
  • a-1 in k, for all nonzero a in k.

For example, the field of rational numbers is a subfield of the field of real numbers which is itself a subfield of the complex numbers. A finite field has the the integers mod p as a subfield.

See also field extension.

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