Does field have subtraction and division Most definitions  (e.g. this http://mathworld.wolfram.com/Field.html) of a field I have seen define two binary operations: addition and multiplication and then axioms on them.
However, recently I came to a doc which stated field has subtraction and division also.
I am confused since as I said most definitions only define two operations: + and *.
How come subtraction or division is also defined?
Where are their rules specified?
 A: It is common to define division as $a / b := a* b^{-1}$ for $b\neq 0$ and substraction as $a - b := a + (-b)$.
A: If $(K,+,\cdot )$ is a field, then $(K,+)$ is also an abelian group, so that the inverse of $x$ is $-x$, so we have subtraction. Similarly, since $(K^{\ast},\cdot)$ is a group, the inverse of $x\neq 0$ is $1/x$, so we have also division.
A: As noted, you can define subtraction and division in terms of the more common definitions of "field".  So, for most purposes, there is no reason to add them.  But why would someone add those extra operations?  Perhaps for certain things done in model theory, it makes a difference what are the primitive operations, and what are only defined operations.
I could also imagine some work on computational efficiency, where subtraction and (more importantly) division are built-in operations of our computing model, and thus count as only 1 operation when we count the number of operations required in a computation.
So, the question is: what doc was that with this alternate definition?
