Compare your test solutions to the online solutions (“Archive” tab has these under the test date). If you have questions about grading, you can contact me with a pic of your problem, or set a time to chat (calendar link on the canvas main page).
Retakes for 2nd Test: 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 7 or higher on your retake problem, you will do a written replacement problem.
If you got 6 or lower, you can schedule to meet in person for a retake. If you want to do this, use the calendar link on the canvas main page for appointments (15 minute slot).
What are the units of $\mathbb{F}_7[x]$? (This is a concept check kind of question.)
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 $x^2 + 2x + 2$?
Consider the ring $\mathbb{F}_3[x]/(x^2+2x+2)$. Reminder, this means the rules are: $3=0$ and $x^2+2x+2 = 0$. You might find it more helpful to write the second rule as $x^2 = x + 1$.
Multiply out and simplify $(x+1)x$.
Multiply out and simplify $(x+1)(x+2)$.
Check your answer against my answer on the #daily-collaboration channel of discord. If things aren’t working, ask for help on discord from me or your peers.
Write out a list of all of the finitely many elements of the ring we are working in.
Write out an addition table for these elements.
Write out a multiplication table for these elements.
On the #daily-collaboration channel of discord, put up a flashcard quiz question: that is, give a problem (to add or multiply two elements) and then put its answer as a spoiler (surrounded by double bars), so someone else can use their addition/multiplication tables to check their answer to your quiz question.
Use your tables to check someone else’s quiz question. In this way, we will probably/hopefully quickly diagnose any problems with the computations and work out the bugs so everyone is getting good at it!
If you make an error in your quiz question, you can use the discord edit functionality to fix it. Eventually we’ll have a bunch of great quiz questions up there to check your work against!
Find the inverse of $x$ in the finite field $\mathbb{F}_7[x]/(x^2+1)$. Remember, this is just like the analogous problem for the integers. It will require an extended euclidean algorithm computation in a polynomial ring.
The multiplication table for the finite field $\mathbb{F}_2[x]/(x^2+x+1)$ can be found in the overleaf notes under “Finite Fields”. Or you can make your own by hand. You can use the multiplication table to do the following problems.
Which elements of this field are invertible? List them. (This is a concept check question.)
Compute the multiplicative order of $x$ in this field.
Compute the multiplicative order of $x+1$.
What is a multiplicative generator for this field (something whose powers generate all invertible elements, like a primitive root)?