Thread: Fields

1. 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?
Any advice will be helpfull.

Thanks

2. 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?
Any advice will be helpfull.

Thanks
The structure of a finite field of order 4 is shown in here. It is $\displaystyle \mathbb{Z}/2\mathbb{Z}[\alpha]/(\alpha^2 + \alpha + 1 )$. Thus, a finite field of order 4 is $\displaystyle \{0, 1, \alpha, \alpha + 1\}$, where $\displaystyle \alpha^2 + \alpha + 1 = 0$. Now you can compute the multiplicative inverse of $\displaystyle \alpha$ and $\displaystyle \alpha + 1$. Note that -1=1 in $\displaystyle \mathbb{Z}/2\mathbb{Z}$.

The structure of the above is given by the fact that
"For any prime power $\displaystyle q=p^n$, $\displaystyle F_q$ is the splitting field of the polynomial $\displaystyle f(T) = T^q - T$ over $\displaystyle F_p$".