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Biography Emmy's father, Max Noether, was descended from a family of wholesale traders in Germany. He had been paralyzed by poliomyelitis at the age of fourteen. He regained mobility, but one leg remained affected. Largely self-taught, he was awarded a doctoratefrom the University of Heidelberg in 1868. After teaching there for seven years, he took a position in the Bavarian city of Erlangen, where he met and married Ida Amalia Kaufmann, the daughter of a prosperous merchant.[8][9][10][11] Max Noether's mathematical contributions were to algebraic geometry mainly, following in the footsteps of Alfred Clebsch. His best known results are the Brill–Noether theorem and the residue, or AF+BG theorem; several other theorems are associated with him, including Max Noether's theorem.
Contributions to mathematics and physics • First and foremost Noether is remembered by mathematicians as an algebraist and for her work in topology. Physicists appreciate her best for her famous theorem because of its far-ranging consequences for theoretical physics and dynamic systems. She showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways.[86][23] Her friend and colleague Hermann Weyl described her scholarly output in three epochs: • Emmy Noether's scientific production fell into three clearly distinct epochs:(1) the period of relative dependence, 1907–1919;(2) the investigations grouped around the general theory of ideals 1920–1926; • (3) the study of the non-commutative algebras, their representations by linear transformations, and their application to the study of commutative number fields and their arithmetics. • —Weyl 1935 • In the first epoch (1907–19), Noether dealt primarily with differential and algebraic invariants, beginning with her dissertation under Paul Gordan. Her mathematical horizons broadened, and her work became more general and abstract, as she became acquainted with the work of David Hilbert, through close interactions with a successor to Gordan, Ernst Sigismund Fischer. After moving to Göttingen in 1915, she produced her seminal work for physics, the two Noether's theorems. • In the second epoch (1920–26), Noether devoted herself to developing the theory of mathematical rings.[87] • In the third epoch (1927–35), Noether focused on noncommutative algebra, linear transformations, and commutative number fields.[88]
First epoch (1908–19) Invariant theory is concerned with expressions that remain constant (invariant) under a group of transformations. As an everyday example, if a rigid yardstick is rotated, the coordinates (x, y, z) of its endpoints change, but its length L given by the formula L2 = Δx2 + Δy2 + Δz2 remains the same. Invariant theory was an active area of research in the later nineteenth century, prompted in part by Felix Klein's Erlangen program, according to which different types of geometry should be characterized by their invariants under transformations, e.g., the cross-ratio of projective geometry. The archetypal example of an invariant is the discriminantB2 − 4AC of a binary quadratic form Ax2 + Bxy + Cy2. This is called an invariant because it is unchanged by linear substitutions x→ax + by,y→cx + dy with determinant ad − bc = 1. These substitutions form the special linear groupSL2. (There are no invariants under the general linear group of all invertible linear transformations because these transformations can be multiplication by a scaling factor. To remedy this, classical invariant theory also considered relative invariants, which were forms invariant up to a scale factor.) One can ask for all polynomials in A,B, and C that are unchanged by the action of SL2; these are called the invariants of binary quadratic forms, and turn out to be the polynomials in the discriminant. More generally, one can ask for the invariants of homogeneous polynomials A0xry0 + ... + Arx0yr of higher degree, which will be certain polynomials in the coefficients A0, ..., Ar, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.
Table 2 from Noether's dissertation [92] on invariant theory. This table collects 202 of the 331 invariants of ternary biquadratic forms. These forms are graded in two variables x and u. The horizontal direction of the table lists the invariants with increasing grades in x, while the vertical direction lists them with increasing grades in u.
Galoistheory • Galois theory concerns transformations of number fields that permute the roots of an equation. Consider a polynomial equation of a variable x of degreen, in which the coefficients are drawn from some ground field, which might be, for example, the field of real numbers,rational numbers, or the integersmodulo 7. There may or may not be choices of x, which make this polynomial evaluate to zero. Such choices, if they exist, are called roots. If the polynomial is x2 + 1 and the field is the real numbers, then the polynomial has no roots, because any choice of x makes the polynomial greater than or equal to one. If the field is extended, however, then the polynomial may gain roots, and if it is extended enough, then it always has a number of roots equal to its degree. Continuing the previous example, if the field is enlarged to the complex numbers, then the polynomial gains two roots, i and −i, where i is the imaginary unit, that is,i 2 = −1. More generally, the extension field in which a polynomial can be factored into its roots is known as the splitting field of the polynomial.
Second epoch (1920–26) • In this epoch, Noether became famous for her deft use of ascending (Teilerkettensatz) or descending (Vielfachenkettensatz) chain conditions. A sequence of non-emptysubsetsA1, A2, A3, etc. of a setS is usually said to be ascending, if each is a subset of the next • Conversely, a sequence of subsets of S is called descending if each contains the next subset: • A chain becomes constant after a finite number of steps if there is an n such that for all m ≥ n. A collection of subsets of a given set satisfies the ascending chain condition if any ascending sequence becomes constant after a finite number of steps. It satisfies the descending chain condition if any descending sequence becomes constant after a finite number of steps.
Eliminationtheory • In 1923–24, Noether applied her ideal theory to elimination theory—in a formulation that she attributed to her student, Kurt Hentzelt—showing that fundamental theorems about the factorization of polynomials could be carried over directly.[108][109][110]Traditionally,elimination theory is concerned with eliminating one or more variables from a system of polynomial equations, usually by the method ofresultants. For illustration, the system of equations often can be written in the form of a matrix M (missing the variable x) times a vectorv (having only different powers of x) equaling the zero vector, M•v = 0. Hence, the determinant of the matrix M must be zero, providing a new equation in which the variable x has been eliminated.
Third epoch (1927–35) • Hypercomplex numbers and representation theory • Noncommutativealgebra
Death • In April 1935 doctors discovered a tumor in Noether'spelvis. Worried about complications from surgery, they ordered two days of bed rest first. During the operation they discovered an ovarian cyst "the size of a large cantaloupe. Two smaller tumors in her uterus appeared to be benign and were not removed, to avoid prolonging surgery. For three days she appeared to convalesce normally, and she recovered quickly from a circulatory collapse on the fourth. On 14 April she fell unconscious, her temperature soared to 109 °F (42.8 °C), and she died. "[I]t is not easy to say what had occurred in Dr. Noether", one of the physicians wrote. "It is possible that there was some form of unusual and virulent infection, which struck the base of the brain where the heat centers are supposed to be located. • A few days after Noether's death her friends and associates at Bryn Mawr held a small memorial service at College President Park's house. Hermann Weyl and Richard Brauer traveled from Princeton and spoke with Wheeler and Taussky about their departed colleague. In the months which followed, written tributes began to appear around the globe: Albert Einstein joined Van der Waerden, Weyl, and PavelAlexandrov in paying their respects. Her body was cremated and the ashes interred under the walkway around the cloisters of the M. Carey Thomas Library at Bryn Mawr.
