For Friday:

1. Tests were returned in class.  I forgot to print the solutions, so they are on canvas.  Please compare the solutions to your test and figure out what went wrong.
2. Retakes:  You may retake one of the 10 point problems in Part B that you attempted.   The written retakes will be posted on canvas (for both Module 2 and 3) as soon as I can manage.
   1. 1. If you got 6 or higher on your retake problem, you will do a written replacement problem.
      2. If you got 5 or lower, you can schedule to meet in person for a retake.  If you want to do this, please email me.
3. What are the units of $\mathbb{F}_7[x]$?  (This is a concept check kind of question.)
4. Do two polynomial long divisions:
   1. 1. In $\mathbb{F}_2[x]$, divide $x^4 + x + 1$ by $x^2 + x + 1$.
      2. In $\mathbb{F}_5[x]$, divide $x^5 + 3$ by $x^2 + 2$.
5. Determine whether $x^2 + x + 1$ is irreducible in $\mathbb{F}_3[x]$.  (This may require an exhaustive search method; you may use Sage if you like, but it’s not that bad by hand if you are efficient/thoughtful on the method.)  What about $x^2 + 2x + 1$?
6. Working in $\mathbb{F}_3[x]$, do the Euclidean algorithm on $x^4+x^3 + x^2  + 2x + 1$ and $x^2 + 2$.  What is the gcd?
7. Challenge:  can you build a finite field with 8 elements?