So far, there is no reason to excise the equivalence class [1 : 0] from our tentative construction of the rational numbers. Since [1:0] is likened to, we may think of (and, indeed, any quotient of a nonzero quantity and zero) as ∞o. However, we might like for our construction of the rational numbers to enjoy the arithmetic attributes of a field, Is our current construction of the rational numbers (including ∞ := [10]), and with binary operations addition and multiplication (as we've defined them) a field (as per the “Classic definition” written on Wikipedia)? (Provably) verify as many axioms as possible, and provide explicit counterexamples to any axioms which are violated. If any of the axioms defining a field are violated due to the inclusion of ∞ = [1 : 0], is the problem remedied by simply removing ∞ := [1 : 0]?

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