Finite Fields – more Models
The story starts here.
———————————————————————–
X: My roommate suggests another model for GF(2).
G: Glad to know. Tell me about it.
X: It is based on binary arithmetic used in computers.
G: What are computers?
X: Don’t you use a computer to surf the Net?
G: I surf in the Net. You see, I’ve logged off from your real world long time ago.
X: Oh yes. Computers are the machines we use to connect to the Net.
G: Ah, those stops of the Net that take and send packets! I always get shocks there. How do computers work?
X: Computers process information by electronic bits — that’s why you get the shocks.
G: What is a bit?
X: “Bit” stands for for a “Binary digit“, a digit in the binary system.
G: I know the binary system — it is based on the number 2.
X: Similar to the base10 system with 10 digits, the base2 system has only two digits: 0 and 1.
G: I see, the same two symbols of GF(2)!
X: Instead of counting in bunches of 10 in decimal system, binary system counts in bunches of 2.
Counting  Decimal  Binary  Base3  Base4  Base5  Note 

zero  0  0  0  0  0  all have zero 
one  1  1  1  1  1  all have one 
two  2  10  2  2  2  run out of symbols in Base2, carry to next bunch 
three  3  11  10  3  3  run out of symbols in Base3, carry to next bunch 
four  4  100  11  10  4  run out of symbols in Base4, carry to next bunch 
five  5  101  12  11  10  run out of symbols in Base5, carry to next bunch 
six  6  110  20  12  11  
seven  7  111  21  13  12  
eight  8  1000  22  20  13  
nine  9  1001  100  21  14  
ten  10  1010  101  22  20  finally run out of symbols in Base10, carry to next bunch 
G: In binary arithmetic, once the digit reaches 2 it is promoted as carry to the next position: 1+1=10 base 2.
X: Yes. 10 base 2 = 1×2+0 = 2.
G: How do these relate to GF(2)?
X: Assume an electronic component with only 1 bit. It won’ have room to store the number 2.
G: It will be forced to drop the carry bit, and store only the remaining bit, which is 0.
X: Thus giving 1+1=0 in base2 carryfree arithmetic!
G: Excellent! Shall we draw up the computations in bits and have a look?
X: I can do that in my head. The 8 easy computations are:
Bit Computations 



G: So what are the dictionaries to make this a model of GF(2)?
X: The dictionaries for the symbols and operations are:
Bit Model of GF(2) 



G: The check is simple: the bit and the symbol are the same. This Bit model for GF(2) is verified correct. Well Done!
X: Now we have two models for GF(2): the Parity model and the Bit model.
G: Excuse my eavesdropping, but this is child’s play!
G: Who’s talking?
G: Johann Carl Friedrich Gauss (17771855), from Germany.
G: Ah, the Prince of Mathematics!
G: And you mention my name in your last letter.
G: Indeed. But master, why do you consider our models for GF(2) child’s play?
G: How do you check the parity of a number?
G: Check whether it is divisible by 2, of course.
G: And the possible remainders of division by 2 are?
G: 0 and 1. Ohoh.
G: And you, how do you convert a number from decimal to binary?
X: By counting in bunches of 2, which is successive division by 2 whenever the amount exceeds 2.
G: For 1 bit, you’ll perform only one division by 2, and the possible results?
X: 0 and 1. Ohoh.
G: So you see, both your models for GF(2) are nothing but my Modulo 2 model of GF(2).
X: Your model?
G: Haven’t you read my book?
X: Which book?
G: Your classic Disquisitiones Arithmeticae (Investigations on Arithmetic) of 1801, I’ve read that.
G: I wrote that at the age of 21. The first chapter is on modular arithmetic.
X: What is modular arithmetic?
G: Fix a number n, called the modulo, then perform add and multiply of numbers as usual, but any result will be divided by the modulo n, keeping only the remainder. I even invent a notation for that.
G: x+y ≡ z mod n is x+y, then keep only the remainder z of division by n. Similar expressions for other arithmetic operations.
X: Let me try an example. For modulo n=3, 2+2=4 ≡ 1 mod 3 because 4 divided by 3 gives remainder 1.
G: When the modulo n=2, there are only 2 remainders: 0 mod 2 and 1 mod 2.
G: In modul0 2 arithmetic, the 8 easy computations are:
Modulo 2 arithmetic 



X: Let me check: for modulo 2, 1+1=2 ≡ 0 mod 2.
G: Hence the Modulo 2 model of GF(2) is:
Modulo 2 (Z_{2}) Model of GF(2) 



G: Again the modulo 2 remainders and symbols match. This Modulo 2 model of GF(2) is verified correct.
G: This is such a natural model for GF(2). How come I miss that?
G: You didn’t miss that, it’s always been in your head after reading my book.
X: I think that’s why you said I can treat the symbols as numbers if I wanted to.
G: The modulo 2 system is denoted by Z_{2}, and your Parity model is implicitly Z_{2}.
X: What is Z?
G: Zahlen in German, or Numbers in English. The modulo is the subscript.
G: Nice meeting you, Gauss. It’s dinner time, shall we find a place to eat and discuss my last letter?
G: Glad to. Let me find a good place using my mobile app.
B: Don’t go. I have another model of GF(2)!
E: Me, too! I have yet another model of GF(2).
G: Long time no see!
G: Shall we continue after dinner?
X: Arrrh — I’m going to log back on after dinner, don’t start without me!
G: Sure.
———————————————————————–
Z: Are you staying for dinner? We have Chinese takeaway tonight.
X: No. I’m heading back to AlGBar right after dinner.
Z: What’s the hurry?
X: The prince and his friends are waiting.
Z: The real Prince? Or the artist formerly known as Prince?
X: No and no, I’ll explain later. My instant noodles are ready.
———————————————————————–
Next