In modern math, the concept of a field establishes addition and multiplication within its structure. We are not free to redefine those without abandoning a boatload of things that depend on their definition.
Division is not inherent to field theory, but rather an operation defined by convention.
It seems like you're fixating on the most common convention, but as Hilel points out, there is no reason we have to adopt this convention in all situations.
In modern math, the concept of a field establishes addition and multiplication within its structure. We are not free to redefine those without abandoning a boatload of things that depend on their definition.
Division is not inherent to field theory, but rather an operation defined by convention.
It seems like you're fixating on the most common convention, but as Hilel points out, there is no reason we have to adopt this convention in all situations.