{"id":22205,"date":"2013-03-06T02:40:42","date_gmt":"2013-03-06T06:40:42","guid":{"rendered":"http:\/\/mathgradblog.williams.edu\/?p=22205"},"modified":"2014-07-22T14:36:52","modified_gmt":"2014-07-22T19:36:52","slug":"spring-break-mathematics","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2013\/03\/06\/spring-break-mathematics\/","title":{"rendered":"Spring Break Mathematics"},"content":{"rendered":"

Our spring break at Channel Islands begins March 17.\u00a0 During my break, I will explore the topic of quadratic number rings and Euclidean Domains.\u00a0 I have already begun assembling and reading literature. Essentially, a Euclidean ring is a domain \"R\" with a map \"\psi\u00a0 such that\u00a0 \"\forall,\u00a0 \"\exists such that<\/p>\n

\"\alpha<\/p>\n

where \"r or \"\psi(r).\u00a0 My Algebraic Number Theory textbook contained the following two references: 1. Samuel. “About Euclidean Rings”. \u00a0Journal of Algebra<\/em>; 1970<\/p>\n

2. Clark. “A Quadratic Field Which is Euclidean but not Norm Euclidean”.\u00a0 Manuscripta Mathematica<\/em>; 1994.<\/p>\n

The first reference talks about Euclidean rings where \"psi\" is not the absolute value of the norm usually defined on the ring.\u00a0 For example, \"\mathbb{Z}\" is a Euclidean ring, but a map \"\psi\" exists such that for \"n\in, \"\psi(n).<\/p>\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.\u00a0 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!\u00a0 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).\u00a0 But there are other real quadratic number rings with unique factorization.\u00a0 Thus, there would be examples of these number rings that are Euclidean, where \"psi\" is not given by the field norm.\u00a0 The number ring<\/p>\n

\"\mathbb{Z}\left[\frac{1}{2}<\/p>\n

was shown in reference 2 to be such an example.\u00a0 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.<\/p>\n

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.\u00a0 Is anybody else planning some independent research or study for spring break?<\/p>\n

\n

<\/div>","protected":false},"excerpt":{"rendered":"

Our spring break at Channel Islands begins March 17.\u00a0 During my break, I will explore the topic of quadratic number rings and Euclidean Domains.\u00a0 I have already begun assembling and reading literature. Essentially, a Euclidean ring is a domain with … Continue reading →<\/span><\/a><\/p>\n

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