Eisenstein

Eisenstein is a surname of German origin, derived from the words for iron (Eisen) and stone (Stein). It is associated with several notable individuals and mathematical concepts.

Sergei Eisenstein (1898–1948) was a Soviet film director and theorist whose work helped shape modern cinema. He is best known for pioneering montage techniques and theoretical writings on film form. His influential films include Battleship Potemkin (1925), October (1927), and Alexander Nevsky (1938). Eisenstein’s ideas about the collision of shots and intellectual montage influenced filmmakers and film theory around the world.

Gotthold Eisenstein (1823–1852) was a German mathematician who contributed to number theory and the theory of modular forms. He formulated results now associated with Eisenstein’s irreducibility criterion for polynomials and helped develop concepts that later bore his name in the study of series and algebraic integers.

In mathematics, several concepts bear the name Eisenstein, named after Gotthold Eisenstein. The Eisenstein integers form a lattice in the complex plane, consisting of numbers of the form a + bω where a and b are integers and ω is a primitive cube root of unity (ω = e^{2πi/3}). The associated norm is N(a + bω) = a^2 − ab + b^2. The Eisenstein primes are primes in this ring. Relatedly, Eisenstein series are certain modular forms that arise in number theory, and the Eisenstein criterion provides a test for the irreducibility of polynomials.

The surname is thus linked to influential figures in cinema and mathematics, with enduring legacies in theory and practice.