  The goal of this paper is two-fold: we generalize the arithmetic Chern-Simons
theory over totally imaginary number fields studied in [Kim15, CKK+16] to
arbitrary number fields (with real places) and provide new examples of
non-trivial arithmetic Chern-Simons invariant with coefficient
$\mathbb{Z}/n\mathbb{Z}$ $(n \geq 2)$ associated to a non-abelian gauge group.
The main idea for the generalization is to use cohomology with compact support
(see [Mil06]) to deal with real places. Before the results of this paper,
non-trivial examples were limited to some non-abelian gauge group with
coefficient $\mathbb{Z}/2\mathbb{Z}$ in [CKK+16] and the abelian cyclic gauge
group with coefficient $\mathbb{Z}/n\mathbb{Z}$ in [BCG+18]. Our non-trivial
examples (with non-abelian gauge group and general coefficient
$\mathbb{Z}/n\mathbb{Z}$) will be given by a simple twisting argument based on
examples of [BCG+18].
