From Why start at x, y, z

The word "field" has two unrelated meanings in mathematics. In algebra, it refers to a nontrivial commutative ring in which every nonzero element is a unit, while in differential geometry and theoretical physics, it refers to a function from a manifold to a vector space. This ambiguity does not exist in some languages like french and german, where fields in the former sense are referred to as "bodies" (french: corps, german: Körper).