Finite Fields – many Models
The story starts here.
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G: Let the first one introduce himself.
B: George Boole (18151864), from England. Did you read my book?
X: Which book?
G: An Investigation of the Laws of Thought of 1854, a classic. I read it just before I logged off.
G: You have another model for GF(2)?
B: Yes, and it isn’t based on numbers.
G: Spell out the details, please.
B: It is based on simple sentences, those you can tell immediately whether it is true or false.
G: This is straight from your book!
B: I also consider simple ways to combine such sentences, only the words and, or, called connectives.
G: That’s two symbols, two operations. Interesting.
B: The connectives can join the simple sentences in 8 possible ways:
Sentence Computations 



X: Exactly how do you compute with sentences?
G: Think logically. I’m here and you’re here: true or false? I’m here or I’m there: true or false?
X: I see. When connected by and, both parts must be true to have true finally. When connected by or, both parts must be false to be false finally.
G: That’s Boole’s Laws of Thoughts!
B: And it gives a model of GF(2) if we use this dictionary:
Boole’s proposed model of GF(2) 



X: That’s brilliant! A symbolic model for a symbolic system!
G: Not quite. This model is wrong!
G: I spot the error: in your model: true or true is true. Translate by the dictionary: 1+1=1, which is not correct in GF(2).
B: Oops! How embarrassing! I made a boolean error!
G: That’s the lesson. We need to check carefully to ensure all dictionary translations are perfect.
B: Wait a second. I can fix my error!
G: Don’t make the same mistake twice.
B: No, I won’t. Instead of or, use exclusive or, shortened to xor. I just redo all the computations:
Sentence Computations 



X: What is xor, or exclusive or?
G: It means either this or that, but not both. Tomorrow either the sun will shine or rain will fall. If tomorrow is foggy or a rainbow appears, my prediction is wrong.
B: And the Boole model of GF(2) is:
Boole model of GF(2) 



G: That’s an impressive quick fix!
G: There are further check … it’s all OK, I declare the Boole model of GF(2) is verified correct.
X: Did you just invent a logical operation to fit into the model?
B: No. It’s all in my book. Using eitheror to join sentences is rather common.
G: Now the second model. Your name please.
E: Euclid of Alexandria (300 BC) from Greece. Excuse me for my dates as I’ve lost count.
G,G,B: (bowing) We all read your book.
X: Which book?
G,G,B: (angry looks) Ssh!
E: My classic text Elements was the bible of geometry for more than 20 centuries, until the rise of modern math.
G: Your method of doing math is still being used in all math textbooks.
E: I try to catch up with modern math, so I’ve been doing a bit of abstract algebra lately.
G: I suppose you have a model of GF(2) based on geometry?
E: Take a nonsymmetric alphabet, say the letter E in my name. There are two ways you can play with it: either leave it, or reflect it to ?.
B: See, Euclid is using my xor!
E: The first task, which is doing nothing, I call it identity. The second task, which is flipping over, I call it reverse.
B: I can see that.
E: Given two tasks, you can either do both of them, or just do the first one and ignoring the second one.
B: Another xor. First is diligent, while the second is lazy.
E: There are 8 ways to combine the tasks with the doboth or doone operations:
Task Computations 



E: This gives a model of GF(2) using the following dictionary:
Euclid’s proposed model of GF(2) 



X: Another impressive feat!
G: Look carefully ¡X this model is wrong!
G: If your Doone means to do the first one, then the Doone table has an error: assume the verticals denote the first, we should have Doone(reverse,identity)=reverse, not identity as shown.
E: Oops, I may not be so good in algebra!
B: We do have to check thoroughly,
E: But I think I can fix this ….. Yes, I can. Instead of Doone meaning to do the first one, let’s change it to do the easy one, assuming the task reverse is harder than the task identity, which is doing nothing.
B: This gives the revised 8 computations:
Task Computations 



E: And with this dictionary we surely have a model of GF(2):
Euclid model of GF(2) 



G: Let me do further check …. they’re OK. I declare the Euclid model of GF(2) verified correct.
G: This Euclid model inspire me to have still another model of GF(2).
X: Another model?!
G: Sure. This one is based on numbers, but not modulo 2.
G: Let hear the details.
G: Take the two numbers +1 and –1. Symbolic + is numerically multiply, and symbolic × is take the maximum.
G: This is very interesting. Let me compute:
Numeric Computations 



B: And use this dictionary for the model of GF(2):
Gauss model of GF(2) 



G: Further checks are OK. This is indeed Gauss model of GF(2).
X: How to get this model from Euclid model?
G: When I see that reversing twice give identity, I recall (–1)(–1)=+1. Euclid used the easier task to pick the identity result in his multiplication, so I try the lesser of –1 and +1. It doesn’t work, but reverting to the bigger of –1 and +1 works.
G: Very good indeed. This inspires me to yet another model of GF(2)!
X: Still more?
G: Yes. Take the symbols 0 and 1 as numbers, but let + to mean difference, and × to mean minimum.
X: What is difference?
G: That’s subtraction without sign: always taking the smaller from the bigger.
X: Oh, I see. Let me do the computations:
Numeric Computations 



G: To match GF(2), the dictionaries for symbols and operations are:
Galois model of GF(2) 



G: That’s excellent.
G: Further checks are OK in this case. This is my Galois model of GF(2).
B: It is thrilling to see so many models of GF(2).
X: Is there a most suitable one among the many choices?
G: That’s a good question. It’s too late. We’ll call it a day.
X: Yes, next time.
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Y: This is way too late.
X: I learn too many models of GF(2) in one day. I’m exhausted.
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