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. \(a\cdot1=1\cdot a=a\)
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.