Foundations

A field $(\mathbb{F},+,\cdot )$ is defined by:

1. Associativity of addition. $a+(b+c)=(a+b)+c$
2. Associativity of multiplication. $a(bc)=(ab)c$
3. Commutativity of addition. $a+b=b+a$
4. Commutativity of multiplication. $ab=ba$
5. Additive Identity. $a+0=0+a=a$
6. Multiplicative Identity. $a\cdot 1=1\cdot a=a$
7. Additive Inverse. $a+(-a)=0$
8. Multiplicative Inverse. $a\cdot a^{-1}=1$
9. Distributivity of Multiplication over Addition. $a(b+c)=ab+ac$

An ordered field satisfies:

1. Exactly one of the following is the case:
   1. $x=y$
   2. \(x
   3. \(y
2. Transitivity. If \(x
3. If \(x
4. If \(x

A complete field satisfies:

Every nonempty subset $A$ of an ordered field $\mathbb{F}$ that is bounded above has a least upper bound.