In the space of square matrices, we characterize row-generated subspaces, on which the determinant is an irreducible polynomial. As a corollary, we characterize square systems of polynomial equations with indeterminate coefficients, whose discriminant is an irreducible hypersurface. This resolves a conjecture of Esterov, and, in a sequel paper, leads to a complete description of components and codimensions for discriminants of square systems of equations.