Let α be a root of the polynomial $f(x) = x^2 + x + 1$.
(a) Construct the field F of 4 elements by adjoining a root α over Z/2. (Write down all elements of F).
(b) Find the inverse of α+1 in F.
I was a little confused by the question as I was not sure if I had to find the actual complex number equal to root α, or if I could just keep using α. I assumed the latter and this was my approach for a):
As Z/2 = {0,1}, then adjoining α to Z/2 to form a field F gives F = ${0, 1, α, α + 1}$.
Then, for b):
Consider $α(α+1) = α^2 + α$. As we know $α^2 + α + 1 = 0$, then $α^2 + α = -1$, congruent to 1 (mod 2). Thus, inverse is α.
Is this approach correct?