What if we don't define 0 as 0, but as "everything which is not this thing"? Strictly philosophically, this is the only thing which makes sense. A thing is only a thing, _nothing_ else. Doing so would enable division by zero, while keeping the field axioms.

Example, $a$ is any algebreic number (including ones divided by this definition of zero). The set of numbers $D=\forall x(x \in \mathbb{A}\land x \ne a) = 0_a$ is the set of all numbers which isn't $a$. In other words, $D$ is 0, in the perspective of $a$.

Calculating $\frac{a}{(D=0_a)}$ would then become every a infinitely large set of numbers. But the key point is that division by zero actually would be defined.

Now I'm just an engineer though, so I might take shortcuts which aren't allowed.