September 10th, 2026

A field $\bF$ is a set together with two operations, addition and multiplication as depicted below, that obeys a list of axioms

1. Commutativity. $x + y = y + x$ and $xy = yx$.

2. Associativity. $(x + y) + z = x + (y + z)$ and $(xy)z = x(yz)$.

3. Distributivity. $x(y + z) = xy + yz$.

4. Identity. There is some $0 \in \bF$ (resp. $1 \in \bF$) such that $a + 0 = a$ (resp. $a \cdot 1 = a$) for every $a \in \bF$.

5. Inverse. For every $a \in \bF$, there is some unique element $-a \in \bF$ (resp. $a^{-1} \in \bF$) such that $a + (-a) = 0$ (resp. $aa^{-1} = 1$). In the case of multiplication, we do not consider $0$ when checking for inverse.

6. Nondegeneracy. $1 \ne 0$.

The field operations, by way of definition, automatically give rise to closure: that, for any $x, y \in \bF$, we have that $x + y \in \bF$ and $x \cdot y \in \bF$.

Let $\bF$ be a field. For any $a, b \in \bF$, we say $a = b$ if they are the same element in the set. We may also introduce a symbol $<$ for comparison. A field $\bF$ is said to be totally ordered if it obeys the following axioms.

1. Trichotomy. For any $x, y \in \bF$, one and only one of these statements hold: $x = y$, $x > y$, $x < y$.

2. Transitivity. If $x < y$ and $y < z$, then $x < z$.

3. Monotonicity. If $x < y$, then $x + z < y + z$ for any $z \in \bF$. Also, $xz < yz$ if $z > 0$.

Suppose $\bF$ is an ordered field. If $S \subseteq \bF$, we say $b$ is an upper bound for $S$ if $x \le b$ for every $x \in S$. We say that $b$ is a least upper bound for $S$ if, of all upper bounds on $S$, $b$ is less than or equal to all other upper bounds. A field $\bF$ is said to be Dedekindcomplete if every subset of $\bF$ that is nonempty and bounded above has a least upper bound.

$\bR$ is the unique ordered and Dedekind complete field.