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A Theorem on Rings

October 16, 2012

***I meant to write this for Ada Lovelace Day in celebration of  “Women in Science and Tech” on 10/16/2012.

Anyone who reads Abstract Algebra book (which I’m going to shorten to just “algebra books”) will know van der Waerden, not to confuse with van der Monde whose name is synonymous with a certain matrix below in Linear Algebra:

This van-der-Waerden guy probably wrote his book in the 30’s but it is probably the oldest Math book still being read by many, many graduate students and advance undergraduate students.  I read his book, didn’t like his style because I hated seeing bad fonts and prints.

But this blog is not about this Dutch mathematician Bartel Leendert van der Waerden.  It’s about his teacher Dr. Noether.  Emmy Noether as she is known today, has stigmatism (a speech Impediment) when she was young.  She showed great talent as a young girl and was one of the only two women in a German  university of about a thousand students.  She diligently wrote a dissertation on Ternary Biquadratic
Forms and latter described her own thesis as “crap”.  I think she was not really a person of low self-esteem, in fact, she was such a perfectionist that she later extend her work from 3 variables to general n variables.

Basically during those time when she extend her work, she was teaching in University of Erlangen without pay.  Then she published papers extending and applying Hilbert’s method to fields of rational functions.  And the invariants of finite groups.  That lead to her
engagement with Abstract Algebra.

I must admit, I don’t know her for a long time in my mathematical life/career.  All I knew was this theorem called “Lasker-Noether Theorem” from some Commutative Algebra book.  I looked at this theorem and went through the proof many, many times, and see it in many different algebra books.  After looking at how it was proved, I always gasped in awe, how something so beautiful could be discovered and proved so elegantly.

Years later, I read in books about Albert Einstein praising Noether and her Noether theorem as one of the most important mathematical theorems ever proved in guiding the development of Modern physics.  Later I found it was not the same as the Lasker-Noether Theorem that I knew very well.  Instead it was something in Physics that I was familiar with but never knew was a famous theorem.  It was describing that if Lagrangian or Hamiltonian was the way to go in Mechanics, then something has to be preserved.  I like the part of Physics where Newton’s second Law was proved and there was no mention of F=ma.

To me, Noether’s idea of using “ideals” in a mathematical ring which is analoguous (but of a different category) to subgroups in a group, is brillant and new things on ascending chain condition and ideals as modules of a ring is “revolutionary”.  van der Waerden described the originality of Noether as “absolute beyond comparison”.

Emmy Noether lived frugally as she was denied pay for her work for a long time.  Even after being paid a salary, she continued to live a simple and modest life. Below are some quotes about her in books written about her:

“wholly engrossed in a discussion of mathematics, ‘gesticulated wildly’ as she ate and spilled her food constantly and wiped it off from her dress, completely unperturbed”

“Appearance-conscious students cringed as she retrieved the handkerchief from her blouse and ignored the increasing disarray of her hair during a lecture. Two female students once approached her during a break in a two-hour class to express their concern, but they were unable to break through the energetic mathematics discussion she was having with other students.”  (Dick, Auguste (1981), Emmy Noether: 1882–1935, Boston: Birkhäuser, ISBN3-7643-3019-8. )


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