Results 1 to 3 of 3

Math Help - Understanding an example about fields

  1. #1
    Newbie
    Joined
    Nov 2008
    Posts
    4

    Understanding an example about fields

    Hi,
    in the first week of lectures we were given the example;

    "F
    = field with two elements = {0,1}
    Where 0 + 0 = 1 + 1 = 0, 0 + 1 = 1 + 0 = 1, 1*0 = 0*1 = 0*0 = 0 and 1*1 = 1.
    In fact, there is a finite field with p elements whenever p is a prime number."

    How can you say that 0+0=1+1=0?
    Doesnt that mean 0=1?
    If so, whats the additive identity here, as isn't it meant to be unique?

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,399
    Thanks
    1327
    Quote Originally Posted by Roland View Post
    Hi,
    in the first week of lectures we were given the example;

    "F
    = field with two elements = {0,1}
    Where 0 + 0 = 1 + 1 = 0, 0 + 1 = 1 + 0 = 1, 1*0 = 0*1 = 0*0 = 0 and 1*1 = 1.
    In fact, there is a finite field with p elements whenever p is a prime number."

    How can you say that 0+0=1+1=0?
    Doesnt that mean 0=1?
    If so, whats the additive identity here, as isn't it meant to be unique?

    No, it doesn't mean that 0= 1. It simply means that the additive inverse of 0 is 0 and the additive inverse of 1 is 1. From the rules 0+ 0= 0, 1+ 0= 1, and 0+ 1= 1 you have that 0 is the additive identity: it added to any other member of the set is that member. 1 is NOT the additive identity because 1+ 1= 0, not 1.
    1 is, of course, the multiplicative identity.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2008
    Posts
    130
    Quote Originally Posted by Roland View Post
    Hi,
    in the first week of lectures we were given the example;

    "F
    = field with two elements = {0,1}
    Where 0 + 0 = 1 + 1 = 0, 0 + 1 = 1 + 0 = 1, 1*0 = 0*1 = 0*0 = 0 and 1*1 = 1.
    In fact, there is a finite field with p elements whenever p is a prime number."

    How can you say that 0+0=1+1=0?
    Doesnt that mean 0=1?
    If so, whats the additive identity here, as isn't it meant to be unique?

    Think of the additive part of finite fields as clock arithmetic, from 0 to 24.

    zero hours past zero o'clock is zero o'clock.
    twelve hours past twelve o'clock is zero o'clock.

    zero hours is the additive identity.

    Clock arithmetic does not form a field, but the problem comes when we try to define multiplication. The addition intuition is still fine. If you know what a group is, this is an example of one.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] trouble understanding a theorem on finite fields.
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: July 13th 2011, 11:41 AM
  2. vector fields - graphing gradient fields
    Posted in the Calculus Forum
    Replies: 0
    Last Post: March 20th 2010, 05:53 PM
  3. Fields
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: August 1st 2008, 12:35 AM
  4. Fields
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 3rd 2008, 10:09 AM
  5. Extension fields / splitting fields proof...
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: December 19th 2007, 07:29 AM

Search Tags


/mathhelpforum @mathhelpforum