Addition table for a 4 elements field Why is this addition table good,
\begin{matrix}
\boldsymbol{\textbf{}+} & \mathbf{0} & \boldsymbol{\textbf{}1} & \textbf{a} &\textbf{ b}\\ 
\boldsymbol{\textbf{}0} & 0 & 1 & a & b\\ 
\boldsymbol{\textbf{}1} & 1 & 0 & b & a\\ 
\boldsymbol{\textbf{} a} & a & b & 0 & 1\\ 
\boldsymbol{\textbf{} b} &b & a & 1 & 0
\end{matrix}
and this one isn't, what makes it not work?
\begin{matrix}
\boldsymbol{\textbf{}+} & \mathbf{0} & \boldsymbol{\textbf{}1} & \textbf{a} &\textbf{ b}\\ 
\boldsymbol{\textbf{}0} & 0 & 1 & a & b\\ 
\boldsymbol{\textbf{}1} & 1 & a & b & 0\\ 
\boldsymbol{\textbf{} a} & a & b & 0 & 1\\ 
\boldsymbol{\textbf{} b} &b & 0 & 1 & a
\end{matrix}
It's clear that $0$ and $a$ changes places in the second table but I can't find an example that refutes any of the addition axioms.

If you have only one operation, it is difficult to speak about field. But, it is well-known that:
1) there exists exactly two groups (up to isomorphism) with 4 elements: one is ${\mathbb Z}/2{\mathbb Z}\times{\mathbb Z}/2{\mathbb Z}$ (the first table) and the other one is ${\mathbb Z}/4{\mathbb Z}$ (the second table)
2) there exists exaclty one field (up to isomorphism) with 4 elements, and it is isomorphic to ${\mathbb Z}/2{\mathbb Z}\times{\mathbb Z}/2{\mathbb Z}$
