The integral trace form of cyclic extensions of odd prime degree

Abstract

Let $L/\mathbb {Q}$ be a cyclic extension of degree~$p$, where $p$ is an odd unramified prime in $L/\mathbb {Q}$. An explicit description of the integral trace form $Tr _{L/\mathbb {Q}}(x^2)|_{\mathfrak O_L}$, where~$\mathfrak O_L$ is the ring of algebraic integers of $L$, is given, and an application to finding the minima of certain algebraic lattices is presented.