  The present paper shows that if $q \in \mathbb P$ or $q = 0$, where $\mathbb
P$ is the set of prime numbers, then there exist characteristic $q$ fields $E
_{q,k}\colon \ k \in \mathbb N$, of Brauer dimension Brd$(E _{q,k}) = k$ and
infinite absolute Brauer $p$-dimensions abrd$_{p}(E _{q,k})$, for all $p \in
\mathbb P$ not dividing $q ^{2} - q$. This ensures that Brd$_{p}(F _{q,k}) =
\infty $, $p \dagger q ^{2} - q$, for every finitely-generated transcendental
extension $F _{q,k}/E _{q,k}$. We also prove that each sequence $a _{p}, b
_{p}$, $p \in \mathbb P$, satisfying the conditions $a _{2} = b _{2}$ and $0
\le b _{p} \le a _{p} \le \infty $, equals the sequence abrd$_{p}(E), {\rm
Brd}_{p}(E)$, $p \in \mathbb P$, for a field $E$ of characteristic zero.
