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April 14th, 2009, 07:17 AM  #1 
Newbie Joined: Apr 2009 Posts: 7 Thanks: 0  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?

April 14th, 2009, 03:49 PM  #2  
Global Moderator Joined: May 2007 Posts: 6,820 Thanks: 722  Re: What is a finite field? Quote:
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.  
April 14th, 2009, 04:24 PM  #3 
Newbie Joined: Apr 2009 Posts: 7 Thanks: 0  Re: What is a finite field?
Ummm... What? Could you explain that to me?

April 14th, 2009, 06:57 PM  #4 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  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 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). 
April 14th, 2009, 07:04 PM  #5 
Newbie Joined: Apr 2009 Posts: 7 Thanks: 0  Re: What is a finite field?
What is affine transformation?

April 14th, 2009, 07:16 PM  #6 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  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. 
April 14th, 2009, 08:42 PM  #7  
Senior Member Joined: Apr 2009 From: Mesa, Arizona Posts: 161 Thanks: 0  Re: What is a finite field? Quote:
 
April 15th, 2009, 05:20 PM  #8  
Global Moderator Joined: May 2007 Posts: 6,820 Thanks: 722  Re: What is a finite field? Quote:
 
April 15th, 2009, 06:47 PM  #9  
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: What is a finite field? Quote:
@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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