## Dissertations, Theses, and Capstone Projects

2-2015

Dissertation

Ph.D.

Mathematics

Victor Kolyvagin

Mathematics

#### Keywords

Gross-Koblitz, Imaginary Quadratic Fields, Norm Equations, Stickelberger's Theorem

#### Abstract

Let $K=\mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic extension of $\mathbb{Q}$. Let $h$ be the class number and $D$ be the discriminant of the field $K$. Assume $p$ is a prime such that $\displaystyle\left(\frac{D}{p}\right)=1$. Then $p$ splits in $K$. The elements of the ring of integers $\mathcal{O}_K$ are of the form $x+\sqrt{-d}y$ if $d\equiv1,2\pmod{4}$ and $\displaystyle x+\frac{1+\sqrt{-d}}{2}y$ if $d\equiv3\pmod{4}$, where $x$ and $y\in \mathbb{Z}$. The norm \\$N_{K/\mathbb{Q}}(x+\sqrt{-d}y)=x^2+dy^2$ and $N_{K/\mathbb{Q}}\left(\displaystyle x+\frac{1+\sqrt{-d}}{2}y\right)=\displaystyle\frac{(2x+y)^2}{4}+\frac{dy^2}{4}$. In this thesis, we find the elements of norm $p^h$ explicitly. We also prove certain congruences for solutions of norm equations.

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