Please ensure you upload a current version of your self-assessment document for daily posts.

For those of you who are still waiting on a paper re-do for the first assessment, we will do it in combination with the second assessment, so keep waiting a couple more days.

Let us consider polynomials in $x$ with coefficients in $\mathbb{Z}/3\mathbb{Z}$. For example, $x + 2x \equiv 0$ and $2x^2 + 2x^2 \equiv x^2$.

Add $2x+2$ to $x^2 + 2x+1$. The result should have coefficients chosen from $0, 1, 2$.

Multiply $x+2$ and $x+1$. The result should have coefficients chosen from $0, 1, 2$.

Recall that we defined $\mathbb{Z}/n\mathbb{Z}$ as the set of integers with the additional “rule” that $n\equiv 0$ so we can remove multiples of $n$. In the same way, consider the set of polynomials in the variable $x$ with coefficients in $\mathbb{Z}/3\mathbb{Z}$ (as in the last question), but “modulo” $x^2+1$. In other words, whenever we see an $x^2+1$, we can remove it. (Hint: this also means whenever we see an $x^2$, we can replace it with $-1 \equiv 2$). For example, $(x+1)(x+1) \equiv x^2 + 2x + 1 \equiv 2+ 2x + 1 \equiv 2x $.

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.

We have defined a new “ring” (number system)! This ring is finite. Write out a list of all of its finitely many elements.

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!