Sun Zi's Theorem (aka The Chinese Remainder Theorem)
Channel: Andrew Misseldine
Date: 2020-11-13

In this video, we present, with proof, the Sun Zi's Theorem, more commonly known in the West as the Chinese Remainder Theorem, from the context of cyclic groups. We show how to solve a system of linear congruences.

This is lecture 28 (part 2/3) of the lecture series offered by Dr. Andrew Misseldine for the course Math 4220 - Abstract Algebra I at Southern Utah University. A transcript of this lecture can be found at Dr. Misseldine's website or through his Google Drive at: 
https://drive.google.com/file/d/1gkPPXm4Qd0E_VqdcSwhe3RR1xeESugyV/view

This lecture is based upon Section 16.5 of Abstract Algebra: Theory and Applications (http://abstract.ups.edu/) by Tom Judson. Please post any questions you might have below in the comment field and Dr. Misseldine (or other commenters) can answer them for you. Please also subscribe for further updates.

It is common tradition in mathematics to name a new result by the author(s) who first discovered it. Unfortunately, our records of history are not always perfect and sometimes the name a theorem is given is instead attributed to someone (or something) else, perhaps to someone who first conjectured or popularized the theorem. Sometimes the original mathematician is only discovered after another name has been coined. As such, renaming a theorem to its proper name is rarely an effort that is truly successful because the traditional name is so entrenched in the history and tradition of mathematics. The name "the Chinese Remainder Theorem" is a victim of this entrenchment. The first historical record of the Chinese Remainder Theorem is found in the Chinese manuscript Sunzi Suanjing (孫子算經) and is believed to be authored by the 3rd century Chinese mathematician Sun Zi.  Very little of Sun Zi other than the authorship of Sunzi Suanjing (for sometime Sun Zi was erroneously identified with the Chinese general Sun Tsu who authored the Art of War). It is not known whether Sunzi Suanjing is an original scholarship of Sun Zi or just a compilation of important mathematical work at the time of Sun Zi, much like a college textbook. It is also not known whether Sunzi Suanjing was written all at once or if Sun Zi added to some previous version drafted by someone else. In the presence of these historical ambiguities, it would be most proper to name the Chinese Remainder Theorem instead "Sun Zi's Theorem" in the absence of a more precise attribution to the original scholar (if it was not, in fact, Sun Zi). The name "Chinese Remainder Theorem" as coined by L E Dickson, an American number theorist , in the early 20th century. He obtained an English translation brought from China by a missionary. Instead of attributing the author of book (who may have been unknown to Dickson at the time), it appears the result is attributed to its Chinese origin.