Hi everyone, sorry I wasn’t feeling well on Friday, so class was cancelled.
As a replacement, I’ve posted a video of the corresponding lecture from last year’s class on canvas. First up, please watch that. It is titled “BONUS: Lecture Math 4440 10-12-2022: Introduction to polynomial arithmetic and finite fields” and appears at the end/bottom of the default listing by creation date in the Media Gallery. (This doesn’t count as your daily post time, it’s lecture-replacement time.) Here are the notes.
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 $x^2 + 2$.
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?
Consider the ring $\mathbb{F}_3[x] / (x+ 1)$.
List all the elements (there should be 3).
Explain in your own words why there are only 3.
Consider the ring $\mathbb{F}_2[x] / (x^2 + x+ 1)$.