# 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 [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

• 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

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

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