  The validity of Korn's first inequality in the fractional setting in bounded
domains has been open. We resolve this problem by proving that in fact Korn's
first inequality holds in the case $ps>1$ for fractional $W^{s,p}_0(\Omega)$
Sobolev fields in open and bounded $C^{1}$-regular domains $\Omega\subset
\mathbb R^n$. Also, in the case $ps<1,$ for any open bounded $C^1$ domain
$\Omega\subset \mathbb R^n$ we construct counterexamples to the inequality,
i.e., Korn's first inequality fails to hold in bounded domains. The proof of
the inequality in the case $ps>1$ follows a standard compactness approach
adopted in the classical case, combined with a Hardy inequality, and a recently
proven Korn second inequality by Mengesha and Scott [\textit{Commun. Math.
Sci.,} Vol. 20, N0. 2, 405--423, 2022]. The counterexamples constructed in the
case $ps<1$ are interpolations of a constant affine rigid motion inside the
domain away from the boundary, and of the zero field close to the boundary.
