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\chapter{EMMY NOETHER} % The first chapter
\label{cha:emmy}

In recent years, there has been a development in mathematical pedagogy to integrate the original ideas of the mathematicians who actually developed the theory, rather than using textbooks that refer to these works. These original sources  oftentimes show why and how problems were solved, and provide insight into the way of thinking of those genius minds. In the preface his book ``Mathematical Expeditions,'' Reinhard Laubenbacher talks about using historical sources in university math classes he taught at New Mexico State University.  The success of those classes was reflected in greater appreciation for the impetus in the development of the topics discussed and in providing the students with a simpler, or less abstract, approach.  This concept of ``learning from the masters'' has led to the growth in interest to translate groundbreaking articles and books written by the most important mathematicians in history.  Laubenbacher suggested that we translate Noether's seminal paper on describing what we now call Noetherian rings and modules. In this work, Emmy Noether describes the precise condition that generalizes principal ideal domains, and thereby forever changing the way mathematicians view rings and modules. 

After an exhaustive search, we discovered that this foundational paper had already been translated by Colin McLarty, professor of mathematics and philosophy at Case Western Reserve University. McLarty is one of the leading experts in the works of Emmy Noether. He translated some of her more important contributions to mathematics and physics and published various articles that highlight her mathematical work. Over correspondence with McLarty, he suggested that there is a rather large number of very influential mathematicians and mathematical historians that would appreciate knowing more about Emmy Noether's contribution to classical invariant theory. This thesis is part of a larger project: translate Noether's dissertation, verify the computations of the algebraic invariants she computed via an archaic symbolic approach, complete the computation of algebraic invariants of fourth degree forms in three variables, and connect the symbolic notation to modern techniques in algebraic invariant theory. Kung and Rota \cite{KungRota} said about this that ``\ldots the covariants of no nontrivial form (except for conics) in three or more variables have been fully classified, not even those of ternary quartic which persuaded Emmy Noether to quit invariant theory.'' This is the project I am presenting in this thesis.

\begin{figure}[htb]
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\includegraphics[scale=0.6]{phoenixashes.jpg}\label{Fig:phoenix}
\caption{Arabian Phoenix}
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In Chapter II of this paper, I will give a quick overview over Emmy Noether's life. In  Chapter III,  I will discuss the major differences between classical and modern invariant theory. In Chapter IV, I will take an  in depth look at classical invariant theory via the well-known binary forms. In Chapter V, I look at modern tools via Sturmfels' book, and an algorithm to compute an invariant ring. Finally, Chapter VI contains the translation of Emmy Noether's doctoral dissertation.


\section{About Emmy Noether}\label{sec:history}

Amalie ``Emmy'' Noether was born on March 23, 1882 in Erlangen, Germany into a Jewish family. Emmy's father was the mathematician Max Noether, who lectured at the University of Erlangen.  Emmy had plans to become a teacher, but instead ended up studying math at the University of Erlangen. She completed her dissertation (which has been translated here) in 1907 under Paul Gordan, and then worked for the university for seven years without getting paid. In 1915, David Hilbert and Felix Klein wanted her to join the mathematics department of the University of G\"ottingen, but again, the university objected to hiring a female. She taught under Hilbert's name for four years and became very well known in the mathematical community. She proved a theorem, now called  ``Noether's Theorem,'' which is one of the most important theorems in mathematics to guide and develop physics. Her theorem is the proof that ``Energy may neither be created nor destroyed'' and other laws of conservation.  Her habilitation was finally approved in 1919, and she was allowed to teach under her own name. In 1920, Noether started working in the area of abstract algebra with Werner Schmeidler. She published papers on theory of ideals and as a result of her work on rings, the term ``Noetherian Rings'' was being used. She worked with B. L. Van der Waerden and was a great influence on the work he published in his book \underline{Moderne Algebra}, which was one of the first central textbook in the algebra. Noether supervised multiple doctoral students, having great influence on their work. Examples of those are Max Deuring, who worked in the area of Arithmetic Geometry, and Wolfgang Krull who advanced the area of Commutative Algebra. Emmy Noether taught at the University of G\"ottingen until 1933, when the Prussian Ministry of Sciences, Art, and Public Education withdrew the right to teach from her. She, and a lot of other Jewish colleagues were forced to find jobs outside of Germany. Through the support of highly influential mathematicians, Emmy Noether was invited to teach at Bryn Mawr University, and also started lecturing at Princeton University in 1934, where Albert Einstein and Hermann Weyl worked. Noether died in 1935 after surgery to remove an ovarian cyst.


\section{Things Other Great Mathematicians Said About Emmy Noether}\label{sec:other}

Solomon Lefschetz from Princton University wrote to Jacob Billikopf at Bryn Mawr University about Emmy Noether:\\
``\ldots she is the holder of a front rank seat in every sense of the word. As the leader of the modern algebra school, she developed in recent Germany the only school worthy of note in the sense, not only of isolated work, but of very distinguished group scientific work. In fact, it is no exaggeration to say that without exception all the better young German mathematicians are her pupils. \ldots''

Norbert Wiener from the Massachusetts Institute of Technology also wrote to Mr Billikopf: \\
`` Leaving all questions of sex aside, she is one of the ten or twelve leading mathematicians of the present generation in the entire world and has founded what is certain to be the most important close-knit group of mathematicians in Germany--the Modern School of Algebraists.''

Albert Einstein writes in his obituary about Emmy Noether: \cite{Roquette}
\\
 ``In the judgment of the most competent living mathematicians, Fr\"aulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.''

Hermann Weyl said the following at Emmy Noether's funeral:\\
``Justifiably proud, for you were a great woman mathematician - I have no reservations in calling you the greatest that history has known. Your work has changed the way we look at algebra, and with your many gothic letters you have left your name written indelibly across its pages. No-one, perhaps, contributed as much as you towards remoulding the axiomatic approach into a powerful research instrument, instead of a mere aid in the logical elucidation of the foundations of mathematics, as it had previously been. Amongst your predecessors in algebra and number theory it was probably Dedekind who came closest.''
\cite{Roquette}


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