Ernst Kummer
Early Life and Education
Ernst Eduard Kummer was born on January 29, 1810, in Sorau, Prussia, now Żary, Poland. He was the son of a physician and showed an early aptitude for mathematics. Kummer attended the University of Halle, where he initially studied theology before switching to mathematics under the influence of Heinrich Ferdinand Scherk. He completed his doctorate in 1831 with a dissertation on the theory of complex numbers.
Academic Career
Kummer began his academic career as a teacher at a gymnasium in Liegnitz, where he taught for over a decade. During this period, he developed many of his mathematical ideas and maintained correspondence with other prominent mathematicians, including Carl Gustav Jacob Jacobi and Carl Friedrich Gauss. In 1842, Kummer was appointed as a professor at the University of Breslau, where he continued his research and teaching.
In 1855, Kummer succeeded Peter Gustav Lejeune Dirichlet as a professor at the University of Berlin, a position he held until his retirement in 1875. His tenure at Berlin was marked by significant contributions to the field of mathematics, particularly in number theory and algebra.
Contributions to Mathematics
Number Theory
Kummer is perhaps best known for his work in number theory, particularly his contributions to the theory of algebraic numbers. He introduced the concept of ideal numbers, which provided a way to address the failure of unique factorization in certain algebraic number fields. This work laid the groundwork for the development of algebraic number theory and influenced subsequent mathematicians such as Richard Dedekind and David Hilbert.
Kummer's interest in number theory was partly motivated by his attempts to prove Fermat's Last Theorem. He made significant progress by proving the theorem for a large class of prime exponents, known as regular primes. His work on Fermat's Last Theorem introduced new techniques and concepts that have become fundamental in modern number theory.
Algebra and Analysis
In addition to his work in number theory, Kummer made important contributions to algebra and analysis. He developed the theory of hypergeometric functions, which are solutions to a particular type of differential equation. Kummer's work on hypergeometric functions expanded the understanding of these functions and their applications in mathematical physics.
Kummer also contributed to the development of invariant theory, a branch of algebra that studies polynomial functions that remain unchanged under transformations. His research in this area influenced the work of later mathematicians, including Felix Klein and Sophus Lie.
Kummer Surfaces
Kummer's name is also associated with a special class of algebraic surfaces known as Kummer surfaces. These surfaces arise from the quotient of a two-dimensional complex torus by the involution that sends each point to its inverse. Kummer surfaces have interesting geometric properties and have been studied extensively in algebraic geometry.
Legacy and Influence
Ernst Kummer's work had a profound impact on the development of mathematics in the 19th century. His introduction of ideal numbers and his contributions to algebraic number theory paved the way for future advancements in the field. Kummer's ideas influenced many prominent mathematicians, including Leopold Kronecker, who was one of his students.
Kummer's work on hypergeometric functions and invariant theory also had lasting effects on the development of these areas. His contributions to mathematics were recognized by his election to various scientific academies, including the Royal Prussian Academy of Sciences and the French Academy of Sciences.
Personal Life
Ernst Kummer married Ottilie Mendelssohn, a member of the prominent Mendelssohn family, in 1840. They had thirteen children, and Kummer was known to be a devoted family man. Despite his demanding academic career, he maintained a balance between his professional and personal life.