My research and investigations

Finite Fields – first Model

The story starts here.
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G: Good day. I’ve found something interesting about GF(2).

X: Didn’t you say that’s all for GF(2)?
G: There is always more.

X: What more can we say about a system with just 2 symbols?
G: The GF(2) we talked about is in the abstract. Now I’ve found a model for GF(2).

X: What is a Model?
G: A model is an interpretation of a system. It assigns meaning to the symbols and operations, thereby verifying all relationships are correct.

X: You mean like a picture?
G: Yeah, a bit. More like an analogy, a model makes an abstract thing meaningful.

X: Sounds magical, also very mysterious.
G: You’ll like this as it involves numbers. For GF(2), I can make a model like this for the symbols:

Symbol Meaning
0 An even number
1 An odd number

X: Can I change the other way round: 0 for odd and 1 for even?
G: I tried, but it won’t work, as you’ll see. Now for the operations:

Operation Meaning
+ Add the two numbers and note the parity of result
× Multiply the two numbers and note the parity of result

X: What is parity?
G: Parity of a number is whether it is even or odd.

X: This looks like a dictionary.
G: Good observation. A dictionary gives meaning to words, which are symbols.

X: What’s the use of this dictionary, or model?
G: Using this model we can check our abstract system GF(2):

Abstract GF(2)
+ 0 1
0 0 1
1 1 0

 

× 0 1
0 0 0
1 0 1

 

X: I thought you said 0 is expelled from multiplication.
G: I swept it under the carpet, but now I pull it out from the carpet.

X: Must be your magic carpet. But why?
G: The system has the relationship 0×a=0 which I cannot hide, although it offends me when working with multiplication.

X: What’s so offending about 0 in multiplication?
G: That’s another rule of Fields that we’ll look at eventually. Today we’ll treat addition and multiplication as equal parts of the structure, so we’ll use the full picture.

X: What equal parts of the structure? You never mention that.
G: The structure of the Field. Add and Multiply are equally important parts.

X: What about Subtract and Divide? I remember you said all four operations must be good.
G: Recall that we’ve made symbolic subtraction work via symbolic addition: a-b=c whenever a=c+b.

X: Ah, that moving symbols around thing!
G: Yes. And we’ve made symbolic division work via symbolic multiplication: a÷b=c whenever a=c×b.

X: So, how to check GF(2) by your model?
G: Simple. Now we can compute our table entries:

Model Computations
Add numbers, note parity Even Odd
Even ? ?
Odd ? ?

 

Multiply numbers, note parity Even Odd
Even ? ?
Odd ? ?

 

X: Can we just use the dictionary to fill in the tables?
G: No! The aim is just the opposite: first compute without using the dictionary, then show that all entries matches GF(2) via the dictionary.

X: You mean we have to work through 8 entries? You said there is a symmetry rule — maybe it can help us to reduce some work.
G: No! Those rules are for the Field. When working with the Model, forget the Field for the moment. Upon completion check with the Field.

X: The actual Model is a lot harder than the abstract Field!
G: Computations are easy, even with 8 of them. You can almost feel the parity of a number. Give them a try.

X: OK. Two evens add to even. Two odds add to even. One odd and one even is still odd.
G: And even makes things even upon multiplication, only odd with odd give odd. So we’re done with the computations:

Model Computations
Add numbers, note parity Even Odd
Even Even Odd
Odd Odd Even

 

Multiply numbers, note parity Even Odd
Even Even Even
Odd Even Odd

 

G: Now check the results of computation with GF(2) via the dictionary. All entries are verified correct.

Abstract GF(2)
Add symbols: + Symbol 0 Symbol 1
Symbol 0 0 1
Symbol 1 1 0

 

Multiply symbols: × Symbol 0 Symbol 1
Symbol 0 0 0
Symbol 1 0 1

 

X: Is there a name for this model?
G: I’ll call it the Parity model of GF(2).

Parity model of GF(2)
Symbol Meaning
0 Even
1 Odd

 

Operation Meaning
+ Add numbers, note parity
× Multiply numbers, note parity

 

X: It’s a good model, giving concrete meaning to abstract symbols.
G: And concrete procedures for abstract symbolic operations.

X: I still can’t see why we don’t cheat by invoking the dictionary for the computations in the first place.
G: When we propose a model, we don’t know if it works. We do all computations purely within the model. If the results are verified correct, then we know the model works. Otherwise we have to reject the model, and try a better one.

X: In that case, what’s the use of model? After this long-winded process, all we know is the model works. It just verifies what’s already known.
G: By its nature, a model reflects exactly the abstract system; nothing more and nothing less. However, by giving meaning to abstract symbols and operations, it explains why the system works, at least in one case.

X: How do you think of a model? It comes from thin air? Who knows what abstract symbols mean?
G: That’s part of the creation process of math, you’ve got to feel the model. You guess the meaning, see if that works.

X: How to guess meanings?
G: For example, GF(2) has two symbols, and I know numbers are of two kinds: Odd and Even. So I try to see if I can match the symbols with parity of numbers.

X: You first build up a dictionary to match symbols?
G: Yes. Then I use the dictionary just for symbols to reveal what the operation tables will look like.

X: Hey, you just said we can’t cheat by invoking the dictionary for the table computations!
G: This is invention. I don’t know what the operations mean: I have to reveal them to guess what they can mean.

X: This is cunning — almost cheating.
G: When I get the dictionary even for operations, I reverse the process: show the dictionaries and compute the tables to verify the model.

X: This is cunningly clever!
G: If a model has been verified, then you can make use of the model to help you reconstruct the abstract system, in case you’ve forgotten a table entry. Generally, the model dictionaries are less demanding to remember than the whole abstract system.

X: So we can use the model to cheat, after all.
G: Only if the model has been verified — that is, someone had done all the computations in the model.

X: Can someone skip the computations and prove once and for all the model works?
G: Now you’re talking math! This can certainly be done, when the theory is sufficiently developed.

X: I think I’m still not of the math-type.
G: Sometimes, quite unexpectedly, we may find two models of the same abstract structure.

X: This is just another dictionary for the symbols and operations. What’s the point of a second verification after the first verification?
G: If that works, it provides another explanation, another good example. It certainly provides further evidence of usefulness of the abstract system.

X: Usefulness, importance and significance of the abstract system.
G: And if the models are from different topics, this can be exciting if this reveals some hidden relationships between the topics.

X: Really? So another model is good?
G: Do you have another model for GF(2)?

X: I think my roommate may have suggested one. I’ll need to talk to him.
G: Fine. Tell me about this tomorrow.
———————————————————————–
Y: How’s today?

X: He changes the subject to models.
Y: As the models parading on the cat-walks?

X: Of course not — as a model giving a concrete picture of an abstract system.
Y: How? How to go from way-up-high to way-down-low?

X: Using a dictionary to give meanings to symbols and operations. That’s an intricate idea.
Y: So you’ve found something interesting in his ideas, at last.

X: Maybe. Explain to me the binary arithmetic of computers again. That can be a model for GF(2).
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