Noether's ( first )  theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law . The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918.  The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function ), from which the system's behavior can be determined by the principle of least action .
From 1908 to 1915, she worked at the Mathematical Institute of Erlangen without pay, and piloted her researches there. Felix Klien and David Hilbert invited Noether to join the mathematics department at the University of Göttingen in 1915. Although she was criticized by many for working at the University, she lectured students for four years under Hilbert’s name. She was given the title ‘Privatdozent’, which permitted her to lecture in 1919, but she was still not paid. In 1922, Noether became an associate professor receiving a menial salary for her service.
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Noether's mathematical work has been divided into three "epochs". In the first (1908–1919), she made contributions to the theories of algebraic invariants and number fields. Her work on differential invariants in the calculus of variations, Noether's theorem, has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics". In the second epoch (1920–1926), she began work that "changed the face of [abstract] algebra". In her classic 1921 paper Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains) Noether developed the theory of ideals in commutative rings into a tool with wide-ranging applications. She made elegant use of the ascending chain condition, and objects satisfying it are named Noetherian in her honor. In the third epoch (1927–1935), she published works on noncommutative algebras and hypercomplex numbers and united the representation theory of groups with the theory of modules and ideals. In addition to her own publications, Noether was generous with her ideas and is credited with several lines of research published by other mathematicians, even in fields far removed from her main work, such as algebraic topology.
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