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A logical approach to abstract algebra

Recent work in constructive mathematics shows that Hilbert's program works for a large part of abstract algebra. Using in an essential way the ideas contained in the classical arguments, we can transform most of the highly abstract proofs of ‘concrete’ statements into elementary proofs. Surprisingly, the arguments we produce are not only elementary but also mathematically clearer, and not necessarily longer. We present an example where the simplification was significant enough to suggest an improved version of a classical theorem. For this we use a general method to transform some logically complex first-order formulae into a geometrical form, which may be interesting in itself.