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April 14th, 2009, 07:17 AM   #1
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What is a finite field?

I have searched all over the internet, but I have not yet found an explanation that I understand. Would someone please explain it to me?
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April 14th, 2009, 03:49 PM   #2
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Re: What is a finite field?

Quote:
Originally Posted by Wolf
I have searched all over the internet, but I have not yet found an explanation that I understand. Would someone please explain it to me?
Which part of the term do you a problem with - field or finite?

Field is a set closed under two binary operations (usually addition and multiplication) and having inverses for all elements, except the addition identity (zero) does not have a multiplicative inverse.
Finte means the set has a finite number of elements.

Simplest examples are sets of integers modulo a prime number.
The smallest such example (mod 2) has 0 and 1 with the following 0x0=0, 0x1=0, 1x1=1, 0+0=0, 0+1=1, 1+1=0. The inverses then are for addition 0 inv =0, 1 inv =1, while for mult. 1 inv =1.
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April 14th, 2009, 04:24 PM   #3
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Re: What is a finite field?

Ummm... What? Could you explain that to me?
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April 14th, 2009, 06:57 PM   #4
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Re: What is a finite field?

Let F be a set with a finite number of elements: e.g. { 0, 1, 2 }. They don't have to be numbers.

Now write out an "addition" table and a "multiplication" table.

Code:
+   0 1 2   x   0 1 2
  /------     /------
0 | 0 1 2   0 | 0 0 0
1 | 1 2 0   1 | 0 1 2
2 | 2 0 1   2 | 0 2 1
The conditions for the combination of set and tables (F, +, ) to be a field are:

CLOSURE. The entries in the tables are all elements of F.
ASSOCIATIVITY. For all a,b,c in F: a + (b + c) = (a + b) + c and a (b c) = (a b) c.
COMMUTATIVITY. For all a and b in F: a + b = b + a and a b = b a
IDENTITIES. There is an equivalent to 0, a "number" which doesn't change any "number" to which it is added.
And there is an equivalent to 1, a "number" which doesn't change any "number" by which it is multiplied.
INVERSES. For any a in F, there are b and c in F such that a + b = 0 and a c = 1. So you can subtract and divide.
DISTRIBUTIVITY. For all a, b and c in F: a (b + c) = (a b) + (a c).

This list is quite restrictive. If any of these conditions aren't met, it ceases to be a field and becomes some other, more general algebraic structure.

It turns out that the number of elements in a finite field has to be a power of a prime number. You could have 8 elements (2x2x2) but not 6 elements (2x3).
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April 14th, 2009, 07:04 PM   #5
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Re: What is a finite field?

What is affine transformation?
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April 14th, 2009, 07:16 PM   #6
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Re: What is a finite field?

In geometry, an affine transformation is a way of altering the "fabric" on which has been drawn a shape or group of shapes. It consists of:

1) a linear transformation (rotation and/or scaling and/or shearing, etc)

followed by

2) a translation (moving rigidly without rotating)

The key is that any possible straight line would remain a straight line, no matter where it was on the fabric.
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April 14th, 2009, 08:42 PM   #7
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Re: What is a finite field?

Quote:
Originally Posted by Wolf
I have searched all over the internet, but I have not yet found an explanation that I understand. Would someone please explain it to me?
Well the way I understand is, a field is a collection of stuffs that have "the friendliest" binary-operators properties known. It become weird as it is mix or miggle with other concepts of mathematics base on set theory and such. For example, it is possible in some situation that one can just lose unity when applying u-substitution for an integral.
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April 15th, 2009, 05:20 PM   #8
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Re: What is a finite field?

Quote:
Originally Posted by Wolf
Ummm... What? Could you explain that to me?
I could respond if I understood what part of my original response was giving you trouble.
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April 15th, 2009, 06:47 PM   #9
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Re: What is a finite field?

Quote:
Originally Posted by MyNameIsVu
Well the way I understand is, a field is a collection of stuffs that have "the friendliest" binary-operators properties known.
Intuitively, yes, but that says nothing useful without saying what "friendly" means.


@Wolf: May I ask where you've found these? You don't seem to have that much of a background, and these are "big" terms. (The rabbit hole just gets deeper. )

Also, are the explanations you've heard so far ok?
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