Fields

A field $F$ is a set equipped with binary operations $+ : F \times F \rightarrow F$ and $\cdot : F \times F \rightarrow F$ called addition and multiplication that satisfy the following list of properties.

$$ \textbf{Commutativity. } x + y = y+x \hspace{10pt} \text{ and } \hspace{10pt} xy = yx \\ \textbf{Distributivity. } x (y+z) = xy+xz \hspace{45pt} \\ \textbf{Additive Identity.} \space \exists 0 \in F :x+0 =x\hspace{64pt} \\ \textbf{Multiplicative Identity. } \exists 1 \in F : 1\cdot x = x \hspace{95pt} \\ \textbf{Additive Inverse. } \exist(-x) \in F : (-x)+x = 0 \hspace{27pt} \\ \textbf{Multiplicative Inverse. } \space \exists \frac{1}{x} \in F: \frac{1}{x}\cdot x =1 \hspace{82pt}

Ordered Fields

An ordered field is a field $F$ equipped with an order $<$ such that

$y < z \implies x + y < x + z$
$x, y > 0 \implies xy > 0$

Sign Rules of Ordered Fields. All the typical sign rules hold for ordered fields.

$$ x> 0 \implies -x < 0 \\ w, x, y, z \geq 0 \wedge x < y \wedge w < z \implies xw < yz \hspace{33pt}\\ 0 < x < y \implies 0 < \frac{1}{y} < \frac{1}{x} \hspace{3pt} $$

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The Real Numbers

There exists an ordered field with the least upper bound property. Moreover, it is unique and contains $\mathbb{Q}$ as an ordered subfield. We denote this field ****$\R$, and give it the name the real numbers.

Properties of $\R$.

Archimedean Property. $\forall x, y \in \R, x> 0, \exists n \in \N _{>0} : nx > y$
$\mathbb{Q}$ is dense in $\R$. $\forall x, y \in \R, x < y, \exists p \in \mathbb{Q} : x < p < y$
$n$-th roots exist in $\R$. $\forall x \in \R {>0}, \forall n \in \N{>0}, \exists! y \in \R_{>0} : y^n = x$

Proofs </aside>