Alexander module of a connected sum (${\mathcal{A}}_{K\#J}\cong {\mathcal{A}}_K\oplus {\mathcal{A}}_J$)

Boundary-connected-summing Seifert surfaces gives a block-diagonal Seifert matrix, because basis curves in the separated summands have zero linking. The Seifert-matrix presentation of the Alexander module is block diagonal too. Thus the Alexander modules form a direct sum and their Alexander polynomials of a knot multiply up to a unit.