Motivated by the prediction of the appearance of atomic multiply charged negative ions (AMCNI) of hydrogen, induced by a linearly polarized laser field, we present an analytical quantum mechanical treatment of the appearance and structure of AMCNI in a linearly polarized field, based on high-frequency Floquet theory (HFFT). For the simplest AMCNI of hydrogen, H2- and H3-, the values of a0 at which the first bound state appears are a0= 1.62x 102 and a0= 1.02x 104, where a0 =I1/2/w2 is the amplitude of the oscillation of a free electron in the field with frequency w and intensity I (unless stated otherwise, we use atomic units throughout this paper). Whereas in vacuum at least one of the electrons of an AMCNI autodetaches, an intense highfrequency field can change the character of the ion dramatically, such that bound states of AMCNI can appear. Due to the interaction with the field, the electrons of the AMCNI oscillate in phase along the polarization axis. This "quiver" motion enables the electrons to be spatially separated over distances of order a0, reducing the repulsive e-e interaction as a0 increases. In other words, for a0 large enough, the field enables a configuration in which the electrons, while widely separated, are bound to one proton. For the prediction of bound states of HN- with N>3, however, a relativistic description or low-frequency theory is required.