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RAMIFICATION IN NUMBER FIELDS GENERATED BY $x^4+p x^2+p$ WITH $p \neq 4+n^2$ A RATIONAL PRIME
Keywords:
discriminant, relative discriminant, ramification, octic fields, indexDOI:
https://doi.org/10.17654/0972555525019Abstract
If $L$ is the splitting field of the polynomial $f(x)=x^4+p x^2+p$ and $p \neq 4+n^2$ is a rational prime, then we have $\operatorname{Gal}(L: \mathbb{Q})=D_8$, where $D_8$ is the dihedral group of order 8 . We calculate the discriminant $\delta_L$ of $L$ without knowing any integral basis of $L$; also, we obtain the explicit ramification of any rational prime $q$ such that $q \mid \delta_L$. For this, we first study the ramification in the intermediate quartic field $K=\mathbb{Q}(\sqrt{p}, \sqrt{p-4})$. We give generators of each ramified rational prime. In some cases, we can make use of Dedekind's Theorem; in other cases, we use the relative extension theory.
Received: November 28, 2024
Accepted: March 7, 2025
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