Page 263 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Traces

Let A be a finite-dimensional commutative K-algebra (with identity), such as a finite extension
field of K or the product ring Kn or even a mixture of the two: a product of finite extensions
of K. To any a 2 A we associate the K-linear map m : A  A which is left multiplication by a:

x  x:

For a; b  A and   K, m = m + m and m = am , so m is a K-linear map.
b
a
a+b
a
aa
a
Definition: For a finite-dimensional K-algebra A, the trace of a  A is the trace of m .
a
That is, the trace of a is tr(m )  K, usually written as Tr A/K (a), so Tr A/K : A  K. The trace from A
a
to K is K-linear, hence identically zero or surjective since K is a one-dimensional K-vector space.
Example: Since m is the identity function, Tr A/K (1) = [A : K].
1
r
Example: Suppose a  A is nilpotent: ar = 0 for some r  1. Then m  0, so m is a nilpotent linear
a
a
transformation. Thus its eigenvalues are all 0, so Tr A/K (a) = 0.
We now consider a finite-dimensional L-algebra A with K a subfield of L such that [L : K] < . We
have finite-dimensional algebras A/L, A/K, and L/K. The next theorem is called the transitivity
of the trace.
Theorem 7: In the above notation, Tr A/K = Tr L/K  Tr A/L . In particular, if a  L, then Tr A/K (a) =
[A : L]Tr L/K (a).
Proof: Let (e1; : : : ; em) be an ordered L-basis of A and (f1; : : : ; fn) be an ordered K-basis
of L. Thus as an ordered K-basis of A we can use
(e f ,....., e f ,....., e f ,....., e f ):
1 1
1 n
m 1
m n
For a  A, let
m n
ij s 
j 
ae  c e , c f  b f ,
ij i
ijrs r
i 1 r 1


for c  L and b  K. Thus a(ef ) = i   r b e f . So
ijrs i r
j s
ijrs
ij
[m ] = (c ), [m ] L/K = (b ), [m ] = ([m ] L/K ):
cij
ijrs
a A/K
a A/L
ij
cij
Thus
(
Tr L/K (Tr A/L (a)) = Tr L/K  c )
ii
i
=  Tr L /K (c )
ii
i
=  b iirr
i r
= Tr A/K (a).
Theorem 8: Let A and B be finite-dimensional K-algebras. For (a; b) in the product ring A × B,
Tr (A×B)/K (a, b) = Tr A/K (a) + Tr B/K (b).
Proof: Let e ,....., e be a K-basis of A and f ,....., f be a K-basis of B. In A × B, e . f = 0. Therefore,
m
i
n
1
1
j
the matrix for multiplication by (a, b), with respect to the K-basis { e f }, is a block-diagonal
i j
 [m ] 0 
a
matrix  0  , whose trace is Tr A/K (a) + Tr B/K (b).
b 
 [m ]
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