# Regular extension

In field theory, a branch of algebra, a field extension is said to be regular if k is algebraically closed in L (i.e., where is the set of elements in L algebraic over k) and L is separable over k, or equivalently, is an integral domain when is the algebraic closure of (that is, to say, 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 is said to be self-regular if 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. Cohn (2003) p.425
3. Fried & Jarden (2008) p.39
4. Cohn (2003) p.426
5. Fried & Jarden (2008) p.44
6. Cohn (2003) p.427