Integral bases and translation

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.

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