# Understanding an example about fields

• Mar 10th 2009, 03:11 AM
Roland
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?

• Mar 10th 2009, 05:18 AM
HallsofIvy
Quote:

Originally Posted by Roland
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.
• Mar 10th 2009, 10:53 AM
robeuler
Quote:

Originally Posted by Roland
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.