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.

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.

If you got 6 or higher on your retake problem, you will do a written replacement problem.

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.

What are the units of $\mathbb{F}_7[x]$? (This is a concept check kind of question.)

Do two polynomial long divisions:

In $\mathbb{F}_2[x]$, divide $x^4 + x + 1$ by $x^2 + x + 1$.

In $\mathbb{F}_5[x]$, divide $x^5 + 3$ by $x^2 + 2$.

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 $x^2 + 2x + 1$?

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?

Challenge: can you build a finite field with 8 elements?