2.5 Subtraction and Division

Unfortunately we are now using the same symbol for two different things, a binary operation on , and a symbol denoting additive inverses.

We also write for . If are both in , then so defines a binary operation on . Also, if , then .

Thus,

- [a)]
- [b)]
- [c)]
- [d)]
- [e)]
- [f)]
- [g)]
- [h)] .

I will now start the practice of using steps in proofs that involve multiple uses of
the associative and commutative laws. For example, I'll write statements such as

with no explanation, because I believe that you recognize that it is correct, and that you can prove it. I'll also write for when I believe that no confusion will result, and I'll use distributive laws like

and

even though we haven't proved them. I will write

and assume that you know (because of our conventions about omitting parentheses; cf. Remark 2.50) that the right side of this means

and you also know (by exercise 2.30) that the parentheses can be rearranged in other sensible orders without changing the value of the expression.

- [a)]
- [b)]
- [c)]
- [d)]
- [e)]

Proof:

Proof: The proof uses the algebraic identity

Since and , we have and hence

Hence if is not a square, then (2.97) has no solutions. If for some , then

- a)
- b)
- c)

There are many other choices we could have made for the field axioms. In [29], Edward Huntington gives eight different sets of axioms that are equivalent to ours. (Two sets of propositions are equivalent if every statement in can be proved using statements in , and every statement in can be proved from statements in .)