On Emmy Noether

"In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant mathematical genius thus far produced since the higher education of women began." (Albert Einstein, New York Times, [Letters to the Editor: 'The Late Emmy Noether: Professor Einstein Writes in Appreciation of a Fellow-Mathematician", 1935)

"[Noether] taught us to think in terms of simple and general algebraic concepts - homomorphic mappings, groups and rings with operators, ideals - and not in cumbersome algebraic computations; and she thereby opened up the path to finding algebraic principles in places where such principles had been obscured by some complicated special situation." (Pavel Alexandrov, 1935)

"The third and last exception to general sterility connects the arithmetic of forms with that other major outgrowth of ancient diophantine analysis, the Gaussian concept of congruence. Dickson in 1907 began the congruencial theory of forms, in which the coefficients of the forms are either natural integers reduced modulo p, p prime, or elements of a Galois field. The linear transformations in the theory, corresponding to those in the classical problem of equivalence, were similarly reduced, and hence modular invariants and covariants were defineable. By 1923 the theory was practically worked out, except for two central difficulties, by Dickson and his pupils. Simplified derivations for some of the results were given (1926) by E. Noether by an application of her methods in abstract algebra." (Eric T Bell, "The Development of Mathematics", 1940)

"The work of Galois and his successors showed that the nature, or explicit definition, of the roots of an algebraic equation is reflected in the structure of the group of the equation for the field of its coefficients. This group can be determined non-tentatively in a finite number of steps, although, as Galois himself emphasized, his theory is not intended to be a practical method for solving equations. But, as stated by Hilbert, the Galois theory and the theory of algebraic numbers have their common root in that of algebraic fields. The last was initiated by Galois, developed by Dedekind and Kronecker in the mid-nineteenth century, refined and extended in the late nineteenth century by Hilbert and others, and finally, in the twentieth century, given new direction by the work of Steinitz in 1910, and in that of E. Noether and her school since 1920." (Eric T Bell, "The Development of Mathematics", 1940)

"Emmy Noether introduced the notion of a representation space - a vector space upon which the elements of the algebra operate as linear transformations, the composition of the linear transformations reflecting the multiplication in the algebra. By doing so she enables us to use our geometric intuition. Her point of view stresses the essential fact about a simple algebra, namely, that it has only one type of irreducible space and that it is faithfully represented by its operation on this space. Wedderburn's statement that the simple algebra is a total matrix algebra over a quasifield is now more understandable. It simply means that all transformations of this space which are linear with respect to a certain quasifield are produced by the algebra. This treatment of algebras may be found in van der Waerden's Moderne Algebra. [...] Recently it has been discovered that this last described treatment of simple algebras is capable of generalization to a far wider class of rings." (Emil Artin, "The influence of J. H. M. Wedderburn on the development of modern algebra", Bulletin of the American Mathematical Society 56 (1), 1950)

"Following [Abraham Fraenkel's] work, Emmy Noether, in 1921, transferred decomposition theorems for ideals in algebraic number fields to those for ideals in arbitrary rings. [...] Noether and her students made other major contributions to ring theory before she turned to a treatment of finite group representations from an ideal-theoretic point of view. [...] Chain conditions had been used since the days of Hölder and Dedekind but were brought to the fore in the 1921 paper [above]. Through Noether's influence [...] algebraic notions were linked to topology in the work of Heinz Hopf and Paul Alexandroff [...]" (Carl B Boyer, "A History of Mathematics", 1968)

"The theory of rings and ideals was put on a more systematic and axiomatic basis by Emmy Noether, one of the few great women mathematicians [...] Many results on rings and ideals were already known [...] but by properly formulating the abstract notions she was able to subsume these results under the abstract theory. Thus she reexpressed Hilbert's basic theorem [...] as follows: A ring of polynomials in any number of variables over a ring of coeffcients that has an identity element and a finite basis, itself has a finite basis. In this reforumulation she made the theory of invariants a part of abstract algebra." (Morris Kline, "Mathematical Thought From Ancient to Modern Times", 1972)

"Noether's theorem demonstrates that wherever there is symmetry in nature, there is also a conservation law, and vice versa. In other words, the symmetries of space and time are not only linked with conservation of energy, momentum, and angular momentum, but each implies the other. Conservation laws are necessary consequences of symmetries, and symmetries necessarily entail conservation laws. [...] The simplicity, power, and depth of Noether's theorem only slowly became apparent. Today, it is an indispensable part of the groundwork of modern physics [...] [with] over a dozen important conservation laws and their associated symmetries×[...]" (Robert P Crease & Charles C Mann, "The Second Creation: Makers of the Revolution in Twentieth-century Physics", 1986)

"A keen mind and infectious enthusiasm for mathematical research made Emmy Noether an effective teacher. Her classroom technique, like her thinking, was strongly conceptual. Rather than simply lecturing, she conducted discussion sessions in which she would explore a topic with her students. [...] Outstanding mathematicians often make their greatest contributions early in their careers. Emmy Noether was an exception: she began to produce her most powerful and creative work around the age of 40 [...]" (Michael Fitzgerald & Ioan James, "The Mind of the Mathematician", 2007)