Spring Break Mathematics

Posted on March 6, 2013 by Casey Barker

Our spring break at Channel Islands begins March 17.  During my break, I will explore the topic of quadratic number rings and Euclidean Domains.  I have already begun assembling and reading literature. Essentially, a Euclidean ring is a domain R with a map \psi : R \rightarrow \mathbb{N}  such that  \forall \alpha, \beta \in R,  \exists q, r \in R such that

\alpha = q \beta + r

where r = 0 or \psi(r) < \psi\left(\beta\right).  My Algebraic Number Theory textbook contained the following two references: 1. Samuel. “About Euclidean Rings”.  Journal of Algebra; 1970

2. Clark. “A Quadratic Field Which is Euclidean but not Norm Euclidean”.  Manuscripta Mathematica; 1994.

The first reference talks about Euclidean rings where psi is not the absolute value of the norm usually defined on the ring.  For example, \mathbb{Z} is a Euclidean ring, but a map \psi exists such that for n\in \mathbb{Z}, \psi(n) \ne |n|.

The second reference re-states a remarkable result from 1973: If the generalized Riemann hypothesis is true, than algebraic number rings with infinitely many units that are Principal Ideal Domains are Euclidean Domains.  Applying this to real quadratic number rings (these rings have infinitely many units), those that have unique factorization are Principal Ideal Domains and would also be Euclidean rings!  There are only sixteen real quadratic number rings that are Euclidean with the map \psi given by the absolute value of the usual norm (the norm of the underlying number field).  But there are other real quadratic number rings with unique factorization.  Thus, there would be examples of these number rings that are Euclidean, where psi is not given by the field norm.  The number ring

\mathbb{Z}\left[\frac{1}{2} + \frac{\sqrt{69}}{2}\right]

was shown in reference 2 to be such an example.  One goal of the above research (not necessarily my goal) is to find a general method of generating the map \psi for real quadratic number rings that have unique factorization, thereby showing that they are in fact Euclidean.

I look forward to exploring the above references and others in more detail during spring break, when I have a lull in my normal classwork, and before I begin my thesis research this summer.  Is anybody else planning some independent research or study for spring break?

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