# Foundations

## Field Axioms

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

- Associativity of addition. \(a+(b+c) = (a+b)+c\)
- Associativity of multiplication. \(a(bc) = (ab)c\)
- Commutativity of addition. \(a+b=b+a\)
- Commutativity of multiplication. \(ab=ba\)
- Additive Identity. \(a+0 = 0+a = a\)
- Multiplicative Identity. \(a\cdot1=1\cdot a=a\)
- Additive Inverse. \(a + (-a) = 0\)
- Multiplicative Inverse. \(a\cdot a^{-1} = 1\)
- Distributivity of Multiplication over Addition. \(a(b+c)=ab+ac\)

## Order Axioms

An ordered field satisfies:

- Exactly one of the following is the case:
- \(x=y\)
- \(x
- \(y

- Transitivity. If \(x
- If \(x
- If \(x

## Completeness Axiom

A complete field satisfies:

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