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perfect field

A field \(k\) is perfect iff

• Every finite extension \(L/k\) is automatically separable, or
• Either \(k\) is characteristic zero, or \(k\) is characteristic \(p\) and the Frobenius \(x\mapsto x^p\) is an automorphism.

Examples

• Of perfect fields:
  • \({\mathbf{Q}}, {\mathbf{R}}, {\mathbf{C}}\), any number field
  • Any finite field \({ \mathbf{F} }_q\)
  • Any algebraically closed field
  • Any algebraic extension of a perfect field.
  • The perfect closure of \({ \mathbf{F} }_p(t)\), i.e. \({ \mathbf{F} }_p(t, t^{1\over p}, t^{1\over p^2}, \cdots)\).
• Of non-perfect fields:
  • \({ \mathbf{F} }_p(t)\), the rational function field in one variable over a finite field
  • \({ \mathbf{F} }_p{\left(\left( t \right)\right) }\), the completion of \({ \mathbf{F} }_p(t)\).