This article discusses neutron scattering measurements on the cuprate, high transition temperature superconductor La2-xSrxCuO4 (LSCO) in an applied magnetic field. LSCO is a type-II superconductor and magnetic flux can penetrate the material via the formation of vorticies. Phase coherent superconductivity characterized by zero resistance is suppressed to the lower field-dependent irreversibility temperature (T-irr(H)) and occurs when the vortices freeze into a lattice. Because superconductivity is destroyed within the vortex cores, an investigation of the vortex state provides information about the ground state that would have appeared had superconductivity not intervened. Our measurements reveal that both optimally doped LSCO (x = 0.16, T-c = 38.5 K) and underdoped LSCO (x = 0.10, T-c = 29 K) have an enhanced antiferromagnetic response in a field. Measurements of the optimally doped system for H = 7.5 T show that inelastic sub-gap spin fluctuations first disappear with the loss of finite resistivity at T-irr, but then reappear at a lower temperature with increased lifetime and correlation length compared to the normal state. In the underdoped system elastic antiferromagnetism develops below T-c in zero field, and is significantly enhanced by application of a magnetic field; phase coherent superconductivity is then established within the anti- ferromagnetic phase at T-irr.