Hello Professor,
Here is another Problem I’m having, It’s from the 2004 exam paper, Q2 ii) :
f(X) = X^4 – p. Let *alpha* be the zero of f.
By making the change of variable g(X) = f(X+1) or otherwise, Show that if p= 3 mod 4 then
{1, *alpha*, *alpha*^2, *alpha*^3}
is an integral basis in Q(*alpha*).
I have calculated the absolute value discriminant of the basis {1, *alpha*, *alpha*^2, *alpha*^3} to equal 64p^3 using N(f’(*alpha*)).
I’m not too sure what method to use, could you perhaps point me in the right direction.
Reply:
The discriminant is incorrect. Be careful of the degree of the number field. In any case, here is a solution.
