Is the Maxwell field local?

Is the classical Maxwell field truly local? This question is raised by several observations, among them the Aharonov-Bohm effect; but the question cannot be answered without a systematic definition and characterization of local subalgebras of observables. This paper reformulates classical field theory in analogy to axiomatic quantum field theory and introduces a precise statement for local independence. (Synonyms for local independence are Einstein causality and principle of maximum signal velocity.) The formal answer of the analysis is: The free Maxwell field does not have local independence. This conclusion is critically discussed.