Factorization of Ideals in Algebraic Number Theory and the Montes Algorithm

Undergraduate Honors Thesis

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Citeable URL: https://scholar.colorado.edu/concern/undergraduate_honors_theses/w37637289

Abstract

Let $f(x) \in \ZZ[x]$ be a monic, irreducible polynomial and $\theta$ a root of $f$. Further, we define the field extension $K := \QQ(\theta)$ and denote its ring of integers by $\mathcal{O}_K$; an essential task in Algebraic Number Theory is to compute the factorization of ideals of $\mathcal{O}_K$ into prime ideals. We begin this thesis by reviewing some basic properties of number fields. Then, we describe a classical result by Dedekind which gives the factorization of the ideal $p\mathcal{O}_K$ for all primes $p$ not dividing the index $\left[\mathcal{O}_K:\ZZ[\theta]\right]$. Finally, we discuss the Montes algorithm which builds off the previous work of Ore in exploiting certain Newton polygons to encode the data needed to create a general factorization algorithm.

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