I would like to construct the field Fp(alpha,beta) where alpha is a root of x^p-x-1 (over Fp[x]) and beta is a root of the polynomial x^p-x-alpha^(p-1) (over Fp(alpha)[x]).
I have tried the following
F0.<x>=GF(p)['x']
f1=x^p-x-1
R1.<alpha1>=F0.quotient(f1)['alpha1']
F1.<x>=Frac(R1)
f2=x^p-x-alpha1^(p-1)
R2.<alpha2>=F1.quotient(f2)['alpha2']
F2.<x>=Frac(R2)
but I think it creates the field F1 well, but it goes wrong for R2... It also feels like there should be a much more straightforward way to do this in sage. What would be the proper way to do this ?
