Page 267 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Add this to both sides to get

'
D
f ( ) f ( )  D (a)  f ( )   D ( )k( ) f ( )  D (a)   D ( )k( )
D
'










2
2
2
1
 '(a)  '(a)

D
'


 f ( ) f (a)  D ( ) .
2
2
 '( )

This proves the formula for a derivation on K() is well-defined. It is left to the reader to check
this really is a derivation.
Example: In contrast with Theorem 12, consider K = F (u) and L = K() where  is a root
p
of X – u  K[X]. This is an inseparable irreducible polynomial over K. The u-derivative on K
p
does not have any extension to a derivation on L. Indeed, suppose the u-derivative on K has an
extension to L, and call it D. Applying D to the equation a = u gives
p
p D() = D(u).
p-1
The left side is 0 since we’re in characteristic p. The right side is 1 since D is the u-derivative on
F (u). This is a contradiction, so D does not exist.
p
Corollary: Let L/K be a finite extension of fields. For any derivation D: K  L and   L
which is separable over K, D has a unique extension to a derivation K()  L. If D(K)  K then
D(K())  K().
Proof: Follow the argument in the proof of Theorem 12, allowing derivations to have values in
L rather than in K(). The formula for D(f()) still turns out to be the same as in (B.1). In
particular, if D(K)  K then the extension of D to a derivation on K() actually takes values in
K().
Self Assessment
1. A ................... ring with no non-zero nilpotent element is called reduced.
(a) associative ring (b) commutative ring
(c) multiplicative ring (d) addition ring
2. An arbitrary field extension ................... is called separable when the ring K  u L is reduced.
(a) L k (b) L/K
-1
(c) K/L -1 (d) (L + K) -1
3. If L/K be a ................... extensions. Then L is separable over K. If and only if any derivation
of K has a unique extension to a derivative of L.
(a) finite (b) infinite
(c) domain (d) split
4. The extension F ( u)/F /4). Since F ( u)  F 2   x /x  u, which then the non-zero
2
2
2
2
nilpotent element ...................
(a) X  u (b) u  X
(c) X  u (d) u  u  x  1
1


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