This activity has three parts.

Part 1

We can define operations \(+\) and \(\cdot\) on the set \(\{0,1,2\}\) according to the following operation tables.  You should read the entry in row \(i\) and column \(j\) as \(i+j\) or \(i \cdot j\) as appropriate.

Addition table
\(+\) 0 1 2
0 0 1 2
1 1 2 0
2 2 0 1


Multiplication table
\(\cdot\) 0 1 2
0 0 0 0
1 0 1 2
2 0 2 1

Show that \( \{0,1,2\} \) with these operations forms a field, which we'll call \( \mathbb{F}_3 \).

Part 2

Is there a field \( \mathbb{F}_4 \) with four elements?

Part 3

Is there a field \( \mathbb{F}_6 \) with six elements?

(I'm not going to publish solutions to these.  I encourage you to work on them and discuss them with your colleagues.  You'll be able to find a solution online if you want to look for one.  These questions are special cases of a more general question (for which \(n\) is there a field with \(n\) elements?), which you'll find explored in the Rings & Modules course in Part A.)

Last modified: Wednesday, 12 October 2022, 10:25 AM