Page 262 - DMTH403_ABSTRACT_ALGEBRA

Unit 27: Separable Extensions

For uniqueness, let D and D be derivations on L which extend the same derivation on K. Since Notes
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D (K)  K and D (K)  K, we have D (F)  F and D (F)  F by Lemma. Then D = D on F since
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1
1
2
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F/K is separable, and D = D on L since L/F is separable.
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When L/K is an algebraic extension of possibly infinite degree, here is the way separability is
defined.
Definition: An algebraic extension L/K is called separable if every finite subextension of L=K is
separable. Equivalently, L=K is separable when every element of L is separable over K.
This definition makes no sense if L/K is not an algebraic extension since a non-algebraic extension
is not the union of its finite subextensions.
Theorem 1 has a problem in the infinite-degree case: there is no natural trace map. However, the
conditions in Theorems 2 and 3 both make sense for a general L/K. (In the case of Theorem 2,
we have to drop the specification of K  K L as a product of copies of K , and just leave the
statement about the tensor product having no non-zero nilpotent elements.) It is left to the
reader to check for an infinite algebraic extension L/K that the conditions of Theorems 2 and 3
match Definition.
The conditions in Theorems 2 and 3 both make sense if L/K is not algebraic, so they could each
potentially be used to define separability of a completely arbitrary field extension. But there is
a problem: for transcendental (that is, non-algebraic) extensions the conditions in Theorems 2
and 3 are no longer equivalent. Indeed, take L = K(u), with u transcendental over K. Then
K  K L = K(u) is a field, so the condition in Theorem 2 is satisfied. However, the zero derivation
on K has more than one extension to K(u): the zero derivation on K(u) and differentiation with
respect to u on K(u).
Definition: A commutative ring with no nonzero nilpotent elements is called reduced.

A domain is reduced, but a more worthwhile example is a product of domains, like F3 × Q[X],
which is not a domain but is reduced.

Definition: An arbitrary field extension L/K is called separable when the ring K  K L is reduced.

Using this definition, in characteristic 0 all field extensions are separable. In characteristic p, any
purely transcendental extension is separable. The condition in Theorem 3, that derivations on
the base field admit unique extensions to a larger field, characterizes not separable field extensions
in general, but separable algebraic field extensions.

A condition equivalent to that in Definition is that F  K L is reduced as F runs over the finite
extensions of K.

The condition that K  K L is reduced makes sense not just for field extensions L/K, but for any
commutative K-algebra. Define an arbitrary commutative K-algebra A to be separable when
the ring K  K A is reduced. This condition is equivalent to A  F being reduced for every finite

extension field F/K.

Example: Let A = K[X]/(f(X)) for any non-constant f(X)  K[X]. The polynomial f(X) need
not be irreducible, so A might not be a field. It is a separable K-algebra precisely when f(X) is a
separable polynomial in K[X].
When [A : K] is finite, an analogue of Theorem 1 can be proved: A is a separable K-algebra if and
only if the trace pairing hx; yi = Tr A/K (xy) from A × A to K is non-degenerate.

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