• Refine Query
  • Source
  • Publication year
  • to
  • Language
  • 1
  • Tagged with
  • 1
  • 1
  • 1
  • 1
  • 1
  • 1
  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

Waring's problem in algebraic number fields

Alnaser, Ala' Jamil January 1900 (has links)
Doctor of Philosophy / Department of Mathematics / Todd E. Cochrane / Let $p$ be an odd prime and $\gamma(k,p^n)$ be the smallest positive integer $s$ such that every integer is a sum of $s$ $k$-th powers $\pmod {p^n}$. We establish $\gamma(k,p^n) \le [k/2]+2$ and $\gamma(k,p^n) \ll \sqrt{k}$ provided that $k$ is not divisible by $(p-1)/2$. Next, let $t=(p-1)/(p-1,k)$, and $q$ be any positive integer. We show that if $\phi(t) \ge q$ then $\gamma(k,p^n) \le c(q) k^{1/q}$ for some constant $c(q)$. These results generalize results known for the case of prime moduli. Next we generalize these results to a number field setting. Let $F$ be a number field, $R$ it's ring of integers and $\mathcal{P}$ a prime ideal in $R$ that lies over a rational prime $p$ with ramification index $e$, degree of inertia $f$ and put $t=(p^f-1)/(p-1,k)$. Let $k=p^rk_1$ with $p\nmid k_1$ and $\gamma(k,\mathcal{P}^n)$ be the smallest integer $s$ such that every algebraic integer in $F$ that can be expressed as a sum of $k$-th powers $\pmod{\mathcal{P}^n}$ is expressible as a sum of $s$ $k$-th powers $\pmod {\mathcal{P}^n}$. We prove for instance that when $p>e+1$ then $\gamma(k,\mathcal{P}^n) \le c(t) p^{nf/ \phi(t)}$. Moreover, if $p>e+1$ we obtain the upper bounds $\ds{\gamma(k,\mathcal{P}^n) \le 2313 \left(\frac{k}{k_1}\right)^{8.44/\log p}+\frac{1}{2}}$ if $f=2$ or $3,$ and $\ds{\gamma(k,\mathcal{P}^n)\le 129 \left(\frac{k}{k_1}\right)^{5.55/ \log p}+\frac{1}{2}}$ if $f\ge4$. We also show that if $\mathcal{P}$ does not ramify then $\ds{\gamma(k,\mathcal{P}^n) \le \frac{17}{2} \left(\frac{k}{k_1}\right)^{2.83/ \log p}+\frac{1}{2}}$ if $f\ge 2$ and $k_1\le p^{f/2}$, and $\ds{\gamma(k,\mathcal{P}^n)\le\left(\frac{f}{p^{f/2-1}}\right)k}$ if $f> 2$ and $k_1> p^{f/2}$.

Page generated in 0.0908 seconds