The Definition of Separable Non-algebraic Extension Fields
Although many important applications of the theory of separable extensions stem from the context of algebraic field extensions, there are important instances in mathematics where it is profitable to study (not necessarily algebraic) separable field extensions.
Let be a field extension and let p be the characteristic exponent of . For any field extension L of k, we write (cf. Tensor product of fields.) Then F is said to be separable over if the following equivalent conditions are met:
- and are linearly disjoint over
- is reduced.
- is reduced for all field extensions L of k.
(In other words, F is separable over k if F is a separable k-algebra.)
Suppose there is some field extension L of k such that is a domain. Then is separable over k if and only if the field of fractions of is separable over L.
An algebraic element of F is said to be separable over if its minimal polynomial is separable. If is an algebraic extension, then the following are equivalent.
- F is separable over k.
- F consists of elements that are separable over k.
- Every subextension of F/k is separable.
- Every finite subextension of F/k is separable.
If is finite extension, then the following are equivalent.
- (i) F is separable over k.
- (ii) where are separable over k.
- (iii) In (ii), one can take
- (iv) For some very large field, there are precisely k-isomorphisms from to .
In the above, (iii) is known as the primitive element theorem.
Fix the algebraic closure, and denote by the set of all elements of that are separable over k. is then separable algebraic over k and any separable algebraic subextension of is contaiend in ; it is called the separable closure of k (inside ). is then purely inseparable over . Put in another way, k is perfect if and only if .
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