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Unit 27: Separable Extensions
K [L:F][F:K] since L=F is separable Notes
K [L:K]
Thus L/K is separable by Theorem 1.2.
Theorem 6: Let L/K be an extension of fields, and L be algebraic over K. Then is separable
over K if and only if any derivation on K has a unique extension to a derivation on K().
Proof: When L is separable over K, Corollary B.10 shows any derivation on K extends
uniquely to a derivation on K().
Now suppose L is inseparable over K. Then (X) = 0, where (X) is the minimal polynomial
of over K. In particular () = 0. We are going to use this vanishing of () to construct a
nonzero derivation on K() which extends the zero derivation on K.
Then the zero derivation on K has two lifts to K(): the zero derivation on K() and this other
derivation we will construct.
Define Z : K() K() by Z(f()) = f(), where f(X) K[X]. Is this well-defined?
Well, if f () = f (), then f (X) f (X) mod (X), say
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f (X) = f (X) + (X)k(X).
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Differentiating both sides with respect to X,
f (X) = f (X) + (X)k(X) + (X)k(X):
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Evaluating both sides at yields f () = f () since () = 0. So Z : K() K() is well-defined.
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It is left to the reader to check Z is a derivation on K(). This derivation kills K, but Z() = 1, so
Z extends the zero derivation on K while not being the zero derivation itself.
The reader can check more generally that when is inseparable over K and K() is arbitrary
the map f() f() is a derivation on K() that extends the zero derivation on K and sends
to . So there are many extensions of the zero derivation on K to K(): one for each element of
K().
We need a lemma to put inseparable extensions into a convenient form for our derivation
constructions later.
Lemma: Let L/K be a finite inseparable field extension. Then there is an L and intermediate
field F such that L = F() and is inseparable over F.
Proof: Inseparable field extensions only occur in positive characteristic. Let p be the characteristic
of K. Necessarily [L : K] > 1. Since L/K is inseparable, there is some L that is inseparable over
K.
Write L = K( ,.... ). We will show by contradiction that some has to be inseparable over K.
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Assume every is separable over K. Then we can treat L/K as a succession of simple field
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extensions as in (2.2), where L = L ( ) with separable over L . By Theorem, any derivation
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on L extends to a derivation on L , so any derivation on K extends to a derivation on L.
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Moreover, this extended derivation on L is unique.To show that, consider two derivations D and
D on L that are equal on K. Since L = K( ) and is separable over K, the proof of Corollary
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B.10 tells us that D and D both send L to L and are equal on L . Now using L in place of K, D and
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D being equal on L implies they are equal on L since L = L ( ) and is separable over L . We
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can keep going like this until we get D = D on L = L. As a special case of this uniqueness, the only
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derivation on L which vanishes on K is the zero derivation on L.
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