Page 259 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes is a product of copies of the field K , so it has no non-zero nilpotent elements. If (X) is inseparable,

then K [X]/((X)) is a product of copies of rings K [Y ]/( Y p m ) with m > 0, which all have
nonzero nilpotents.

Now we consider the structure of K  L when L/K is any finite extension.
K
First assume L/K is separable. By the primitive element theorem, we can write L = K() and 
is separable over K. By the first paragraph of the proof, K  K L K [L:K] since (X) has distinct

linear factors in K .
If L/K is inseparable, then some a  L is inseparable over K. Tensoring the inclusion map
K()  L up to K , we have an inclusion

K  K()  K  L.
K
K
The ring K  K() has a non-zero nilpotent element by the first paragraph of the proof, so
K
K  L does as well.
K
Corollary: The proof of Theorem 2 implies Theorem 4.
Proof: Make a tower of intermediate extensions in L/K as in (2.2). Note K is an algebraic
closure of every field L in the tower. Since
i
K  K L  (K  K L )  L1 L
1
and L = K( ) with  separable over K, the proof of Theorem 2 implies
1
1
1
K  K L  K [L 1 :K]
1
as K -algebras. Therefore

K  K L  K [L 1 :K]  L 1 L  (K  L 1 L) [L 1 :K]
Since L = L ( ,...  ) with each  separable over L , we can run through the same computation
i
1
2
1
r
for K  L 2 L as we did for K  K L, and we get K  L 1 L  (K  L 2 L) [L 2 :L ] , so
1
K  K L  (K  L 2 L) [L 2 :L ][L :K] = (K  L L) [L 2 :K] .
1
1
2
Repeating this enough, in the end we get
K  K L (K  r L L) [L r :K]  K [L:K] .

Corollary: The proof of Theorem 2 implies Theorem 5.
Proof: The field K is an algebraic closure of F and L. Using Theorem 2,
K  L  ( K  F) L
K
K
F
 K [F:K] L since F/K is separable
F
 ( K  L) [F:K]
F

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