  Linear intersection pairs of linear codes have become of interest due to
their nice algebraic properties and wide applications. In this paper, we focus
on linear intersection pairs of cyclic codes over finite fields. Some
properties of cyclotomic cosets in cyclic groups are presented as key tools in
the study of such linear intersection pairs. Characterization and constructions
of two cyclic codes of a fixed intersecting dimension are given in terms of
their generator polynomials and cyclotomic cosets. In some cases, constructions
of two cyclic codes of a fixed intersecting subcode are presented as well.
Based on the theoretical characterization, some numerical examples of linear
intersection pairs of cyclic codes with good parameters are illustrated.
