The OP didn't invent the definition of fields as defined by additional and multiplication, with division defined as the Inverse of multiplication. Even the Wikipedia article on fields (https://en.wikipedia.org/wiki/Field_%28mathematics%29#Defini...) defines them this way. (Not that Wikipedia is an authority on truth, but I find that it's a pretty good indicator of what positions are common/widespread regarding a subject.) It's a bit silly to claim that division is defined as axiomatic "in mathematics" given how fundamental fields, defined as the OP defined them, are to modern number theory and mathematics in general.