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In field theory, a branch of algebra, a field extension $L/k$ is said to be regular if k is algebraically closed in L Template:Clarification needed and L is separable over k, or equivalently, $L\otimes _{k}{\overline {k}}$ is an integral domain when ${\overline {k}}$ is the algebraic closure of $k$ (that is, to say, $L,{\overline {k}}$ are linearly disjoint over k).

Properties

• Regularity is transitive: if F/E and E/K are regular then so is F/K.
• If F/K is regular then so is E/K for any E between F and K.
• The extension L/k is regular if and only if every subfield of L finitely generated over k is regular over k.
• Any extension of an algebraically closed field is regular.
• An extension is regular if and only if it is separable and primary.
• A purely transcendental extension of a field is regular.

Self-regular extension

There is also a similar notion: a field extension $L/k$ is said to be self-regular if $L\otimes _{k}L$ is an integral domain. A self-regular extension is relatively algebraically closed in k. However, a self-regular extension is not necessarily regular.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

References

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• M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese) [1]
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• A. Weil, Foundations of algebraic geometry.

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