  Zagier-Hoffman's conjectures predict the dimension and a basis for the
$\mathbb Q$-vector spaces spanned by $N$th cyclotomic multiple zeta values
(MZV's) of fixed weight where $N$ is a natural number.
  For $N=1$ (MZV's case), half of these conjectures have been solved by the
work of Terasoma, Deligne-Goncharov and Brown with the help of Zagier's
identity. The other half are completely open. For $N=2$ (alternating MZV's
case) and $N=3,4,8$, Deligne-Goncharov and Deligne solved the same half of
these conjectures for $N$th-cyclotomic MZV's. For other values of $N$, no sharp
upper bound on the dimension is known.
  In this paper we completely establish, for all $N$, Zagier-Hoffman's
conjectures for $N$th cyclotomic multiple zeta values in positive
characteristic. By working with the tower of all cyclotomic extensions, we
present a proof that is uniform on $N$ and give an effective algorithm to
express any cyclotomic multiple zeta value in the chosen basis. This
generalizes all previous work on these conjectures for MZV's and alternating
MZV's in positive characteristic.
