  For a closed orientable connected 3-manifold $M$, its complexity
$\boldsymbol{T}(M)$ is defined to be the minimal number of tetrahedra in its
triangulations. Under the assumption that $M$ is prime (but not necessarily
atoroidal), we establish a lower bound for the complexity $\boldsymbol{T}(M)$
in terms of the $\mathbb{Z}_{/2\mathbb{Z}}$-coefficient Thurston norm for
$H^1(M;\mathbb{Z}_{/2\mathbb{Z}})$: (1) for any rank-1 subgroup $\{0,\varphi\}
\leqslant H^1(M;\mathbb{Z}_{/2\mathbb{Z}})$, we have $\boldsymbol{T}(M)
\geqslant 2+2||\varphi||$ unless $M$ is a lens space with
$\boldsymbol{T}(M)=1+2||\varphi||$; (2) for any rank-2 subgroup
$\{0,\varphi_1,\varphi_2,\varphi_3\} \leqslant
H^1(M;\mathbb{Z}_{/2\mathbb{Z}})$, we have $\boldsymbol{T}(M) \geqslant
2+||\varphi_1||+||\varphi_2||+||\varphi_3||$. Under the extra assumption that
$M$ is atoroidal, these inequalities had already been shown by Jaco,
Rubinstein, and Tillmann. Our work here shows that we do not need to require
$M$ to be atoroidal.
