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.


The discriminant is incorrect. Be careful of the degree of the number field. In any case, here is a solution.


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