Page 257 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes
Example: Consider the extension F ( u )/F (u). Since F ( u )  F [X]/(X – u),
2
2
2
2
2
2
2
2
F (u)  2 F (u) F ( u)  F ( u)  F (u)[X]/(X – u) = F (u)[X]/(X – u) ,
2
2
2
2
which has the nonzero nilpotent element X – u .
Theorem 3: Let L/K be a finite extension. Then L is separable over K if and only if any derivation
of K has a unique extension to a derivation of L.
For above two proofs, the reader should be comfortable with the fact that injectivity and
surjectivity of a linear map of vector spaces can be detected after a base extension: a linear
map is injective or surjective if and only if its base extension to a larger field is injective or
surjective.
Each of the three theorems above will be proved and then lead in its own way to proofs of the
following two theorems.
Theorem 4: If L = K(a ,....., a ) and each a is separable over K then every element of L is separable
1
i
r
over K (so L/K is separable).
Theorem 5: Let L/K be a finite extension and F be an intermediate field. If L/F and F/K are
separable then L/K is separable.
We will use our new viewpoints to define separability for arbitrary (possibly non-algebraic)
field extensions.

We want to show L/K is separable if and only if Tr L/K : L  K is not identically 0. The trace map
is either identically 0 or it is onto, since it is K-linear with target K, so another way of putting
Theorem 1 is that we want to show L/K is separable if and only if the trace from L to K is onto.
Proof: We might as well take K to have positive characteristic p, since in characteristic 0 all
finite field extensions are separable and the trace is not identically 0 : TrL (1) = [L : K]  0 in
/K
characteristic 0.
If L/K is separable, by the primitive element theorem we can write L = K() where  is separable
over K. To show the trace is surjective for finite separable extensions, it suffices to prove surjectivity
of the trace map on K()/K when K is any base field and  is separable over K.
If L/K is inseparable, then there must be some a  L which is inseparable over K. Since
Tr L/K = Tr K()/K o Tr L/K() , it success to prove the trace map on K()=K vanishes when  is inseparable
over K.
For both cases of the field extension K()/K ( separable or inseparable over K), let  have
minimal polynomial (X) in K[X]. Write (X) =  (Xpm) where m is as large as possible, so  (X)
is separable. Thus (X) is separable if and only if m = 0.

Let n = deg  = p d, with d = deg  . In K[X],
m
 (X) = (X –  ) ... (X –  )
d
1
for some  ’s, which are all distinct since  (X) is separable. Write   g p i m , so the  ’s are distinct.
i
i
i
Then
m
m
p
p


 (X)   (X ) (X p m  1 ) ...(X p m  d ) (X   1 ) p m ...(X   d ) .

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