\definition{Fields and Field Axioms}
A \textit{field} $\mathbb{F}$ consists of a set $A$ and two operations $+$ and $\times$. \\
As usual, we may abbreviate $a \times b$ as $ab$. \\
The following axioms must be satisfied for any $a, b, c \in \mathbb{F}$:

\vspace{1mm}
\begin{center}
% @{} supresses the space between columns.
% @{=} makes = a column seperator.
\begin{tabular}{l | r@{=}l | r@{=}l}
	\hline
		\multicolumn{1}{|c|}{Name} &
		\multicolumn{2}{c}{$+$} &
		\multicolumn{2}{|c|}{$\times$} \\
	\hline
	Closure			& \multicolumn{2}{c|}{$a+b \in \mathbb{F}$} & \multicolumn{2}{c}{$ab \in \mathbb{F}$} \\
	Associativity	& $(a+b)+c~$&$~a+b+c$ 	& $(ab)c~$&$~a(bc)$ \\
	Commutativity	& $a+b~$&$~b+a$ 		& $ab~$&$~ba$ \\
	Distributivity	& $a(b+c)~$&$~ab + ac$	& \multicolumn{2}{}{} \\
	Identity		& $a+0~$&$~a$			& $1 \times a~$&$~a$ \\
	Inverses		& $a + (-a)~$&$~0$		& $a \times a^{-1}~$&$~1$
\end{tabular}
\end{center}


\problem{}
Show that all fields are groups. \\
Convince yourself that not all groups are fields.

\vfill


\problem{}
Is $\mathbb{Z}$ a field under our usual definitions of $+$ and $\times$? \\
Which axioms does it satisfy, and which does it violate?

\vfill

\problem{}
Verify that $\mathbb{R}$ is a field.
\vfill

\generic{Remark:}
We won't worry too much about fields this week. They simply provide a foundation for \textit{spaces}. \\
As such, you may assume that we are working in $\mathbb{R}$ for the rest of this handout.

\pagebreak