The set of rational numbers is a field: Wrong answers

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To show that a set is a field, we need to check all the axioms of the field CONCRETELY. Just listing the axioms is not a proof.
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I read this on a forum, but I don't understand it, but I think it is a simpler way to show that Q is a field

"R is a field therefore all you need to show is that addition and multiplication of two rational numbers does not take you out of the rationals, because then Q inherits all the other field properties from R"

If only I could know why all those properties are inherited

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