Regular extension

In field theory, a branch of algebra, a field extension L/k is said to be regular if k is algebraically closed in L (i.e., k = \hat k where \hat k is the set of elements in L algebraic over k) 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

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

References

  1. Fried & Jarden (2008) p.38
  2. 2.0 2.1 Cohn (2003) p.425
  3. 3.0 3.1 3.2 Fried & Jarden (2008) p.39
  4. Cohn (2003) p.426
  5. Fried & Jarden (2008) p.44
  6. Cohn (2003) p.427