## My research and investigations

### Finite Fields – first Example

The story starts here.
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X: I was thinking: should I come again? Learning only two symbols per day will take years to learn Finite Fields.
G: That’s nothing. The math community took decades to understand my ideas.

G: Don’t worry. We shall just pick the simple stuff and enjoy.

X: Keep things simple. Recall that I have math-phobia.
G: Promise. Do you recall the building blocks of Finite Fields?

X: Building blocks? Something you said just before goodbye … the two symbols 0 and 1?
G: So you did learn something yesterday.

X: I recall that only because I can’t understand this. Somehow 0 is expelled from multiplication, so start with 1. To me they are trivial numbers — I can’t see how they can be called building blocks.
G: Symbols! 0 and 1 are the first two symbols of any Field!

X: Alright, don’t get mad. You said I can think of abstract symbols as numbers without harming the theory.
G: Did I? Well, that’s true. So try to add and multiply these two symbols — or numbers in your mind.

X: Simple and easy: 0+0=0, 0+1=1, 1+1=2, and 1×1=1 since 0 is taken out from multiplication.
G: We’ve got only two symbols 0 and 1 — there is no symbol 2.

X: Can we stop this silly argument? Why can’t a genius invent a symbol 2 for the number 2?
G: Please calm yourself. We have to play by the rules.

X: What rules?
G: The good rules of Fields.

X: There is something good in all these? I’m listening.
G: OK, the first rule for Fields is this: the result of add and multiply cannot go beyond the system.

X: Remind me what is system.
G: The set of symbols we are playing with.

X: Did you just invent a rule to prevent me from doing 1+1=2?
G: Yes and no. Gauss, Euler and Fermat, among others, had worked in such systems before me, but I’m the first to see that there is a wonderful theory for such systems following from a few good rules.

X: If you say so. Then how are you going to play the game by your own rule? There’s an ugly hole in the addition table:

+ 0 1
0 0 1
1 1 ?

G: Easy. There are only two symbols to choose from: 0 or 1.

X: What is your choice: something+something=something, or something+something=nothing, genius?
G: What do you think? I’ll take the opposite of your choice.

X: Adding something to something of course end up in at least something! I’d say in symbols: 1+1=1.
G: Add can mean anything — it is just a symbolic operation. I’d rather look at patterns. To me, 1+1=0 is pretty:

+ 0 1
0 0 1
1 1 0

X: Pretty? You do math based on pretty or not pretty? And they regard you as genius?
G: Oh yes, I single-handedly invented the theory of symmetry. Want to know?

X: Don’t you scare me with another theory! What’s so pretty about your 1+1=0?
G: With this the addition table is just so symmetrical. Look at the reflection along the diagonals!

X: This is crap. Why don’t you invent a rule for prettiness?
G: Come to think of it, you’re right. The second rule for Fields is: each symbol appears once and only once in rows or columns of tables.

X: That’s a Sudoku rule!
G: What is Sudoku?

X: Never heard of that? The popular puzzle to fill 9×9 squares with digits 1 to 9, once and only once along rows, columns or 3×3 boxes.
G: That’s a wonderful game of symbols, with a more restrictive rule than the one for Fields.

G: I’ll just need one diagonal symmetry. The third rule for Fields is: add and multiply should be symmetric.

X: Exactly what do you mean by being symmetrical?
G: For all symbols a and b in the system, a+b=b+a and a×b=b×a.

X: You’re just making up rules to get along?
G: The rules are in my head. I’ve never bother to spell them out. I guess you help me to bring them out clearly.

X: I just realize that my 1+1=1 also fits into your symmetry rule.
G: But it doesn’t fit into the Sudoku rule — that’s more powerful.

X: I can see that the Sudoku rule is more powerful.
G: I can feel the power of the Sudoku rule. Anyway, here is an example of a Field: arithmetic operations are good:

Field with 0 and 1
+ 0 1
0 0 1
1 1 0

× 1
1 1

X: What’s the big deal? You just jotted down enough rules to make an example with two symbols.
G: This is the smallest Field — a Finite Field with two elements. I shall denote it by GF(2).

X: What’s GF?
G: Galois Field.

X: You name it after yourself?
G: No. The math community just pick my name for these fields when they understand my ideas.

X: You said a Field has add, subtract, multiply and divide all good. I can see add and multiply are good here. How about the others?
G: Ah, I almost forget about them! In GF(2), subtract is the same as add, divide is the same as multiply — so both are very good.

X: What? The math genius says: a-b = a+b, and a÷b = a×b? What kind of math is this?
G: The kind of math in GF(2), that is. It just happens to be so: a-b = b-a for all a, b; and a÷b = b÷a for nonzero a, b.

X: Am I talking to a crazy genius? My math teacher will surely fail you for your crazy subtract and divide.
G: Divide and multiply are symbolic operations. They are supposed to be opposite – one undoing the other. But in GF(2) there is not much you can multiply: only 1×1=1 as 1 is the only nonzero. Therefore a/b or b/a can only be 1÷1=1, same as 1×1=1.

X: But at least there are more symbols for add and subtract, even in GF(2).
G: Just think logically. Whatever add does, subtract will undo it. What do you think 1-1 should be?

X: For numbers that’s easy: 1-1=0.
G: Does it look the same as: 1+1=0 in GF(2)? Surely if adding gives no gain, subtracting will give no loss.

X: But 1 and 0 are symbols, according to you. Can you deduce 1-1=0 as symbols, not numbers?
G: Yes. A symbol equals to itself, hence 1=1, therefore 1-1=0.

X: Hey, you are just moving symbols around!
G: There is a rule for moving symbols around in Fields.

X: That’s the same trick again — you’re just inventing rules to cover yourself.
G: All the rules are in my head, I just use them. You need to be able to move symbols around to undo things:

• a-b=c is equivalent to a=c+b;
• a÷b=c is equivalent to a=c×b.

Indeed, the Persian mathematician al-Khwa-rizmi- (780-850) described this moving symbols around as al-jabr (restoration by balancing), from which the word algebra is derived. His name is the origin of the word algorithm.

X: You’re pulling out history to back you up. What about 0-1 in symbols?
G: Use algebra to move the symbols around: 0=1+1, so 0-1=1 in GF(2).

X: That’s really crazy: -1=1 in GF(2)!
G: That’s crazy but true. That’s what makes subtraction and addition the same in GF(2).

X: Are we done? I’ve heard enough of crazy talks today.
G: That’s all. GF(2) with two symbols 0 and 1 is defined by the fact that 1+1=0, or 1=-1.
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Y: How’s the meeting with the 0-1 genius?

X: He’s crazy.
Y: A crazy genius? Tell me more.

X: Any math teacher will fail a student who asserts that 1=-1.
Y: You mean he can’t tell the difference between positive and negative numbers?

X: He keeps saying that they’re symbols, not numbers. It is as if he can do anything he likes with symbols.
Y: There is something called symbolic computation in computer science. I’m interested. How comes 1=-1?

X: He insists that’s correct in GF(2).
Y: What is GF(2)?

X: Something of his own invention, with just the two symbols 0 and 1 and the special rule 1+1=0.
Y: Moving the +1 from the left side to the right side of equality, surley 1=-1.

X: That’s exactly his crazy way! Isn’t just moving symbols around to do math is something crazy? something not math, or SNM?
Y: No. Moving symbols around is math. The topic of symbolic logic is all about doing logical deductions by moving symbols around. The mathematician David Hilbert (1862-1943) was of the opinion that doing math is like a game of chess: to prove a theorem is to find the correct moves. He even claimed that all math can be reduced to such moving around of symbols, which he called formal methods.

X: Has the world turned crazy?
Z: There was this episode in physics. In 1958 Wolfgang Pauli was presenting his new theory of matter with Niels Bohr and others in the audience. At the end of the lecture Pauli asked Bohr his opinion. Bohr’s reply was, “We are convinced your theory is crazy. The question which divides us is whether it is crazy enough to be true!”

Y: Actually, this is not crazy: it’s true. 1+1=0 is what happens inside computers.

X: I can’t believe this crazy equation has applications!
Y: All modern computers are binary machines: at the lowest level they’re manipulating only two symbols 0 and 1 based on binary arithmetic.

X: What is binary arithmetic?
Y: Just like ordinary arithmetic which is based on the number 10, binary arithmetic is based on the number 2. It works like this ….
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