# Algebraic field

A field is a commutative ring $$(R, +, \times)$$ (henceforth abbreviated simply as $$R$$, with multiplicative identity $$1$$ and additive identity $$0$$) which additionally has the property that every nonzero element has a multiplicative inverse: for every $$r \in R$$ there is $$x \in R$$ such that $$xr = rx = 1$$. Conventionally we insist that a field must have more than one element: equivalently, $$0 \not = 1$$.

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