  The $12$-dimensional Fomin-Kirillov algebra $FK_3$ is defined as the
quadratic algebra with generators $a$, $b$ and $c$ which satisfy the relations
$a^2=b^2=c^2=0$ and $ab+bc+ca=0=ba+cb+ac$. By a result of A. Milinski and H.-J.
Schneider, this algebra is isomorphic to the Nichols algebra associated to the
Yetter-Drinfeld module $V$, over the symmetric group $\mathbb{S}_3$,
corresponding to the conjugacy class of all transpositions and the sign
representation. Exploiting this identification, we compute the cohomology ring
$Ext_{FK_3}^*(\Bbbk,\Bbbk)$, showing that it is a polynomial ring $S[X]$ with
coefficients in the symmetric braided algebra of $V$. As an application we also
compute the cohomology rings of the bosonization $FK_3\#\Bbbk\mathbb{S}_3$ and
of its dual, which are $72$-dimensional ordinary Hopf algebras.
