  We present basic constructions and properties in arithmetic Chern-Simons
theory with finite gauge group along the line of topological quantum field
theory. For a finite set $S$ of finite primes of a number field $k$, we
construct arithmetic analogues of the Chern-Simons 1-cocycle, the
prequantization bundle for a surface and the Chern-Simons functional for a
$3$-manifold. We then construct arithmetic analogues for $k$ and $S$ of the
quantum Hilbert space (space of conformal blocks) and the Dijkgraaf-Witten
partition function in (2+1)-dimensional Chern-Simons TQFT. We show some basic
and functorial properties of those arithmetic analogues. Finally we show
decomposition and gluing formulas for arithmetic Chern-Simons invariants and
arithmetic Dijkgraaf-Witten partition functions.
