We bound the symbol length of elements in the Brauer group of a field
K
K
containing a
C
m
C_m
field (for example any field containing an algebraically closed field or a finite field), and solve the local exponent-index problem for a
C
m
C_m
field
F
F
. In particular, for a
C
m
C_m
field
F
F
, we show that every
F
F
central simple algebra of exponent
p
t
p^t
is similar to the tensor product of at most
len
(
p
t
,
F
)
≤
t
(
p
m
−
1
−
1
)
\operatorname {len}(p^t,F)\leq t(p^{m-1}-1)
symbol algebras of degree
p
t
p^t
. We then use this bound on the symbol length to show that the index of such algebras is bounded by
(
p
t
)
(
p
m
−
1
−
1
)
(p^t)^{(p^{m-1}-1)}
, which in turn gives a bound for any algebra of exponent
n
n
via the primary decomposition. Finally for a field
K
K
containing a
C
m
C_m
field
F
F
, we show that every
F
F
central simple algebra of exponent
p
t
p^t
and degree
p
s
p^s
is similar to the tensor product of at most
len
(
p
t
,
p
s
,
K
)
≤
len
(
p
t
,
L
)
\operatorname {len}(p^t,p^s,K)\leq \operatorname {len}(p^t,L)
symbol algebras of degree
p
t
p^t
, where
L
L
is a
C
m
+
ed
L
(
A
)
+
p
s
−
t
−
1
C_{m+\operatorname {ed}_L(A)+p^{s-t}-1}
field.