Math Department (Thesis/2007-08)

The Gaussian Integers

Under the direction of: Paul Friedman

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The set of complex numbers can be viewed as a two-dimensional vector space over the real numbers with basis $\{1, i=\sqrt{-1}\}$, that is, $${\bf C}=\{\,a\cdot 1 +b\cdot i\mid a, b\in {\bf R}\,\}.$$ As you know, $\bf C$ has both a well-defined operation of addition and of multiplication. Restricting $\bf C$ to the set $G$ of those elements that have integer coefficients, that is, $$G=\{\,a+bi\mid a, b\in {\bf Z}\,\}$$ produces the Gaussian integers. This set has many properties similar to that of the integers. Let's study this set: what are the "primes", what about a division algorithm, what about unique factorization?

This is a one (possibly two)-term thesis topic.

Prerequisite: Math 235 or 221 or permission of instructor.

Math Department web pages Created: 25 Apr 2007 Last modified: Apr 25, 2007 6:05:47 PM Comments to: math@union.edu

Thesis Topics - P. Friedman Thesis Topics for 2007-2008 Thesis Topics - P. Friedman