Section 1
The Galois Field $G F_p$
Show that if $F$ is a field, then $a 0=0$ for any $a$ in $F$. (Look at $a 0+a 0$.)
Show that if $F$ is a field, and $a b=0$, then $a=0$ or $b=0$. (Show that if $a \neq 0$, then $b=0$.) Another way to state this is that if $a$ and $b$ are nonzero, then so is $a b$.
Solve the following system of equations over $G F_5$ :$$\begin{aligned}& 3 x+7 y+6 z=2 \\& 4 x+5 y+3 z=2 \\& 2 x+4 y+5 z=4\end{aligned}$$
Construct the Cayley tables for $G F_{11}$.
Solve the system$$\begin{aligned}& 3 x+9 y+4 z=8 \\& 2 x+3 y+5 z=2 \\& 5 x+4 y+9 z=10\end{aligned}$$over $G F_{11}$.
Construct the Cayley tables for $G F_{13}$.
Construct the Cayley tables for $G F_{17}$.
Explain how to locate $a^{-1}$ in the multiplicative Cayley table for $G F_p$.
Explain why the set of integers modulo 10 under addition and multiplication is not a field.
Explain why the operations on congruence classes $\bmod p,[a]+[b]=[a+b]$ and $[a][b]=[a b]$, are well defined. That is, why does the result not depend on the particular choice of $a$ and $b$ from the congruence class?
Solve the equation $x^2+7 x+10=0$ over $G F_{11}$.
Factor the polynomial $3 x^3+x^2+11 x+1$ over the field $G F_{17}$.