Regular extension

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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 [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).[1][2]

Properties[edit]

  • Regularity is transitive: if F/E and E/K are regular then so is F/K.[3]
  • If F/K is regular then so is E/K for any E between F and K.[3]
  • The extension L/k is regular if and only if every subfield of L finitely generated over k is regular over k.[2]
  • Any extension of an algebraically closed field is regular.[3][4]
  • An extension is regular if and only if it is separable and primary.[5]
  • A purely transcendental extension of a field is regular.

Self-regular extension[edit]

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.[6] However, a self-regular extension is not necessarily regular.[citation needed]

References[edit]

  1. ^ Fried & Jarden (2008) p.38
  2. ^ a b Cohn (2003) p.425
  3. ^ a b c Fried & Jarden (2008) p.39
  4. ^ Cohn (2003) p.426
  5. ^ Fried & Jarden (2008) p.44
  6. ^ Cohn (2003) p.427