Fields

• September 23rd 2009, 05:39 PM
shing
Fields
let F4 = (0,1,a,b) be a field containing 4 elements.
Assume 1+ 1 = 0
Prove that b = a^-1 = a^2 = a + 1

What i did:

if b = a^-1
then b.a = (a^-1).(a)
so ab = 1

How can i prove the whole in equality?
Where do i go from here?

Thanks
• September 23rd 2009, 06:12 PM
aliceinwonderland
Quote:

Originally Posted by shing
let F4 = (0,1,a,b) be a field containing 4 elements.
Assume 1+ 1 = 0
Prove that b = a^-1 = a^2 = a + 1

What i did:

if b = a^-1
then b.a = (a^-1).(a)
so ab = 1

How can i prove the whole in equality?
Where do i go from here?
The structure of a finite field of order 4 is shown in here. It is $\mathbb{Z}/2\mathbb{Z}[\alpha]/(\alpha^2 + \alpha + 1 )$. Thus, a finite field of order 4 is $\{0, 1, \alpha, \alpha + 1\}$, where $\alpha^2 + \alpha + 1 = 0$. Now you can compute the multiplicative inverse of $\alpha$ and $\alpha + 1$. Note that -1=1 in $\mathbb{Z}/2\mathbb{Z}$.
"For any prime power $q=p^n$, $F_q$ is the splitting field of the polynomial $f(T) = T^q - T$ over $F_p$".