Tensor product of irreducible representation and one-dimensional representation is irreducible

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Statement

Suppose G is a group and K is a field. Suppose \rho_1:G \to GL(V_1) is an [[irreducible linear representation] of G over K and \rho_2: G \to GL(V_2) is a one-dimensional linear representation of G over K. Then, the tensor product of linear representations \rho_1 \otimes \rho_2 is an irreducible linear representation of G over K. Also, this tensor product is projectively equivalent to \rho_1.