It is shown that the Brauer factor set
(
c
i
j
k
)
({c_{ijk}})
of a finite-dimensional division algebra of odd degree
n
n
can be chosen such that
c
i
j
i
=
c
i
i
j
=
c
j
i
i
=
1
{c_{iji}} = {c_{iij}} = {c_{jii}} = 1
for all
i
,
j
i,j
and
c
i
j
k
=
c
k
j
i
−
1
{c_{ijk}} = c_{kji}^{ - 1}
. This implies at once the existence of an element
a
≠
0
a \ne 0
with
tr
(
a
)
=
tr
(
a
2
)
=
0
{\text {tr}}(a) = {\text {tr}}({a^2}) = 0
; the coefficients of
x
n
−
1
{x^{n - 1}}
and
x
n
−
2
{x^{n - 2}}
in the characteristic polynomial of
a
a
are thus
0
0
. Also one gets a generic division algebra of degree
n
n
whose center has transcendence degree
n
+
(
n
−
1
)
(
n
−
2
)
/
2
n + (n - 1)(n - 2)/2
, as well as a new (simpler) algebra of generic matrices. Equations are given to determine the cyclicity of these algebras, but they may not be tractable.