Ramification in quartic cyclic number fields $K$ generated by $x^4+px^2+p$

Abstract

If $K$ is the splitting field of the polynomial $f(x)=x^4+px^2+p$ and $p$ is a rational prime of the form $4+n^2$, we give appropriate generators of $K$ to obtain the explicit factorization of the ideal $q{\mathcal O}_K$, where $q$ is a positive rational prime. For this, we calculate the index of these generators and integral basis of certain prime ideals.

Keywords

• ramification
• cyclic quartic field
• discriminant
• index