Immersed in an ionic solution, a network of polyelectrolyte polymers imbibes the solution and swells, resulting in a polyelectrolyte gel. The swelling is reversible, and is regulated by ionic concentrations, mechanical forces, and electric potentials. This paper develops a field theory to couple large deformation and electrochemistry. A specific material model is described, including the effects of stretching the network, mixing the polymers with the solvent and ions, and polarizing the gel. We show that the notion of osmotic pressure in a gel has no experimental significance in general, but acquires a physical interpretation within the specific material model. The theory is used to analyze several phenomena: a gel swells freely in an ionic solution, a gel swells under a constraint, electric double layer at the interface between the gel and the external solution, and swelling of a gel of a small size.