I'm not studying algebra yet, I just happened to notice this and am curious. Mathworld's entry for the field axioms doesn't include closure axioms, but I have seen other authors include closure axioms in the field axioms. Does anyone know why this is or what difference it makes? Can closure be deduced from the other axioms?(adsbygoogle = window.adsbygoogle || []).push({});

Without the closure axioms, could you prove, for instance, that the sum and product of a nonzero rational number and an irrational number are irrational? The only way I know how to prove this is with the closure axioms.

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# Field axioms with or without closure

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