Quasi-algebraically Closed Field - Examples

Examples

  • Any algebraically closed field is quasi-algebraically closed. In fact, any homogeneous polynomial in at least two variables over an algebraically closed field has a non-trivial zero.
  • Any finite field is quasi-algebraically closed by the Chevalley–Warning theorem.
  • Algebraic function fields over algebraically closed fields are quasi-algebraically closed by Tsen's theorem.
  • The maximal unramified extension of a complete field with a discrete valuation and a perfect residue field is quasi-algebraically closed.
  • A complete field with a discrete valuation and an algebraically closed residue field is quasi-algebraically closed by a result of Lang.
  • A pseudo algebraically closed field of characteristic zero is quasi-algebraically closed.

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