Page 261 - DMTH403_ABSTRACT_ALGEBRA
P. 261
Abstract Algebra
Notes Now replace K as base field with K(), over which the s are of course still separable. Then any
i
derivation on K() extends uniquely to a derivation on L. But in the proof of Theorem we saw
there is a non-zero derivation Z on K() that vanishes on K, and an extension of that to a
derivation on L is non-zero on L and is zero on K. We have a contradiction of the uniqueness of
extensions, so in any set of field generators { ,...., }, some must be inseparable in K.
r
1
i
Choose a generating set { ,...., } with as few inseparable elements as possible. At least one i
1
r
is inseparable over K and we may assume that is one of them. Set = and F = K( ,.... )
1
r
r
r-1
(so F = K if r = 1). Then L = F(). We will show by contradiction that must be inseparable
over F, which is the point of the lemma.
Suppose is separable over F. Then is separable over the larger field F( ) since its minimal
p
p
p
polynomial over F( ) divides its minimal polynomial over F. Since is a root of X p
F( )[X], its (separable) minimal polynomial in F( )[X] is a factor of this, so that polynomial
p
p
k
p
p
k
must be X . Therefore, F( ). Taking p -th powers for any k 0, a F(a p k+1 ), so
k
p
F(a ) F(a p k+1 ).
The reverse inclusion is obvious, so F(a ) = F(a p k+1 ) for all k 0. Therefore,
k
p
L = F( ) = F( p k ) = K( 1 ,..., r-1 , pk )
r
k
p
for any k 0. We can pick k so that is separable over K (why?). Then the generating set
p
k
{ 1 ,..., r 1 ,a } has with one less inseparable element among the field generators. This
r
contradicts the choice of generators to have as few members in the list as possible that are
inseparable over K, so has to be inseparable over F.
Proof: Assume L/K is separable, so by the primitive element theorem L = K() where is
separable over K. Any derivation on K can be extended (using Theorem) uniquely to a derivation
on L.
If L/K is inseparable, then Lemma lets us write L = F() with inseparable over F, and
F K. The, by a construction used in the proof of Theorem, f() f() with f(X) F[X] is a
nonzero derivation on L which is zero on F, and thus also zero on the smaller field K. This shows
the zero derivation on K has a non-zero extension (and thus two extensions) to a derivation on L.
Corollary: The proof of Theorem 3 implies Theorem 4.
Proof: Again we consider the tower of field extensions (2.2). Since L = L ( ) and is separable
i-1
i
i
i
over L , the proof shows any derivation on L extends uniquely to a derivation on L . Therefore,
i-1
i-1
i
any derivation on K = L can be extended step-by-step through the tower (2.2) to a derivation on
0
Lr = L. By the argument in the proof of Lemma, this derivation on L is unique.
Lemma: Let L/K be a finite extension and F be an intermediate extension such that F/K is
separable. Then any derivation F L which sends K to K has values in F.
Proof: Pick F, so is separable over K. Now use Corollary B.10 to see the derivation F L
sends to an element of K() F.
Corollary: Theorem 3 implies Theorem 5.
Proof: To prove L/K is separable, we want to show any derivation on K has a unique extension
to a derivation on L. Since F/K is separable, a derivation on K extends to a derivation on F. Since
L/F is separable, a derivation on F extends to a derivation on L.
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