My research and investigations

Finite Fields – no Math

The story starts here.

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X: What is a Finite Field? I don’t want to know the math!
G: Hey, you’re talking to a math genius!

X: OK, just the barely minimum – I’m afraid of math but I’d like to know a little bit of it.
G: In that case, it’s simple: a Finite Field is a Field that’s finite.

X: That’s the answer from a genius?
G: Well, that’s the truthful answer.

X: You are playing with words.
G: I’m playing with symbols.

X: How will you play with just one word: what is finite?
G: Finite is not infinite.

X: I’ll stop talking to you.
G: That’s the logical reply. If you don’t know finite and infinite intuitively, you’d better go home.

X: You simply haven’t answered my original question. What is a Field then?
G: Do you know how to add, subtract, multiply and divide?

X: You mean like 2+3, 5-4, 6×7 and 8÷2?
G: I’m not really thinking of numbers … but numbers are OK.

X: That’s simple and easy. I learnt them long ago — perhaps that’s the only time in my life I like math.
G: Yes, that’s math. A Field is a system within which add, subtract, multiply and divide are all good.

X: Why do you say system? That sounds abstract. Your system has numbers for the arithmetic operations, right?
G: I’m thinking of symbols, not numbers. If that sounds abstract, you can think of a system as a set of numbers — that’ll do no harm to the theory.

X: Wow! The word theory has such a strong smell of math that it scares me! Can you not mention any theory?
G: Alright, no theory. Let me start from the beginning. Can you count?

X: Everybody counts: 1, 2, 3, 4, 5, ….
G: I count from zero: 0, 1, 2, 3, 4, 5, …

X: I count with fingers: 1 finger, 2 fingers, … Nothing has nothing to count.
G: Just close all the fingers to show 0 fingers!

X: I know, I know! I don’t have all day to argue with you.
G: Come to think of it, I count in my head from 0. If I count with fingers, 1, 2, 3 seem better.

X: Hey math genius — you can’t even decide counting from 1 or 0 ?
G: Oh I’ve decided. I’ll denote the set of numbers {0,1,2,3,…} as N, and the set without zero as N* = {1,2,3,…}.

X: The word set worries me. Once my math teacher talked about set theory; at first I understand it, then I don’t.
G: Don’t worry. For me a set is just a collection of things. I’ll use a ∊ S to mean object a is in set S.

X: That’s math! That’s the scary math I’m sooooooo afraid of!
G: That’s just a symbol in the math language, like + for add and × for multiply. Alright, I’ll just say a in S, ok?

X: That’s better. So where were we? Math is really not for me.
G: I was saying that there’ll be two counting sets N and N*, with N including zero and N* excluding zero.

X: I can’t believe I’m talking to a math genius for several minutes just on how to start counting numbers!
G: No, they are symbols. I’m going to play with symbols.

X: Numbers, symbols. Potatoes, po-ta-toes. Who cares?
G: This is important: I’m going to add and multiply with symbols — that can be very different from add and multiply with numbers!

X: Add and multiply symbols, not numbers? Are you out of your mind? Can I say #+%, @*! ?
G: Don’t be rude. Add and multiply are just symbolic operations. This is how Romans add: III + IV = VII.

X: These are Roman symbols for numbers. They are just doing arithmetic on numbers with weird symbols.
G: I shall use numbers as symbols, and perform arithmetic on symbols denoted by familiar-looking numbers.

X: Can’t argue with a genius who must have a twisted mind.
G: You’ll see this is the heart of Finite Fields. Let’s start with the first two symbols: 0 and 1

X: Why not just start with one symbol 0? I know how to add and multiply this symbol: 0+0=0, 0×0=0.
G: That’s not a Field.

X: I thought you’d say “that’s trivial”. Why is it not a Field?
G: You still recall what a Field means?

X: You said something about a Field is all good — I can’t recall all the details.
G: A Field is a system within which add, subtract, multiply and divide are all good.

X: I don’t know about system!
G: A Field is a set of symbols within which add, subtract, multiply and divide are all good.

X: So a system is a set of symbols?
G: Now you know what a system is!

X: But I can add my zero: 0+0=0, multiply my zero: 0×0=0, aren’t they all good?
G: I like zero for adding, not for multiplying. I’ll just acknowledge 0×(anything)=0 and then sweep it under the carpet.

X: Out of sight, out of mind?
G: In fact, from now on I’ll just multiply non-zero symbols. So just a zero will have no multiplication — that’s why it’s not a field.

X: Is this how a genius do math — just depends on whether he likes it or not?
G: No. Symbolic multiplication involving zero is just no good, as you’ll see.

X: I can sense that there is a lot more behind your simple good.
G: Ah, I haven’t define my good — that’s the beginning of the theory.

X: Arrh …. no theory please. If it’s good for you, it’ll be good for me.
G: Good. So let’s start with two symbols: 0 and 1.

X: I feel hungry. Don’t you need to eat something after all this talk?
G: I’ll need to top up my energy level, too. We can continue tomorrow.

X: What have I learned today? Just 0 and 1!
G: They are the building blocks of Fields. See ya.
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Y: What have you learned today?

X: It’s a waste of time — I just learned how to count.
Y: To learn counting from a genius? What’s new?

X: Genius? He can’t even decide whether to count from 1 or 0!
Y: There is a popular rumor in computer science: those who count from 1 prefer Pascal, while those who count from 0 prefer C.

X: Eventually he decided to count from 0 when adding, but to count from 1 when multiplying.
Y: If he were a programmer, he will be mastering both Pascal and C! Quite a genius.

X: This genius can’t explain Finite; his words: Finite is not infinite — what a jerk!
Y: That’s in the same spirit as recursive acronym, very common nowadays. For example, GNU stands for GNU Not Unix.

Z: The Chinese linguist/humorist Lin Yutang (林語堂) was once posed with this question, “What is the definition of an ideal husband?”. His reply, “The husband of an ideal wife”. No one raised the next question, since everyone knew the answer. A burst of laughter followed.
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