  For a 3-manifold $M$ and an acyclic
$\mathit{SL}(2,\mathbb{C})$-representation $\rho$ of its fundamental group, the
$\mathit{SL}(2,\mathbb{C})$-Reidemeister torsion $\tau_\rho(M) \in
\mathbb{C}^\times$ is defined. If there are only finitely many conjugacy
classes of irreducible representations, then the Reidemeister torsions are
known to be algebraic numbers. Furthermore, we prove that the Reidemeister
torsions are not only algebraic numbers but also algebraic integers for most
Seifert fibered spaces and infinitely many hyperbolic 3-manifolds. Also, for a
knot exterior $E(K)$, we discuss the behavior of $\tau_\rho(E(K))$ when the
restriction of $\rho$ to the boundary torus is fixed.
