Ernst Eduard 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 his early education was influenced by his father's scientific background. Kummer attended the University of Halle, where he initially studied theology but soon shifted his focus to mathematics under the influence of the renowned mathematician Heinrich Christian Schumacher. He completed his doctorate in 1831 with a dissertation on the theory of functions, a field that would become central to his later work.
Academic Career
Kummer began his academic career as a teacher at a gymnasium in Liegnitz (now Legnica, Poland) in 1832. During this period, he continued his research in mathematics, focusing on number theory and algebra. His work caught the attention of Carl Gustav Jacob Jacobi, a leading mathematician of the time, who encouraged Kummer to pursue a university position. In 1842, Kummer was appointed as a professor at the University of Breslau (now Wrocław, Poland), where he continued to develop his mathematical theories.
Contributions to Mathematics
Kummer's most significant contributions to mathematics were in the fields of number theory and algebra. He is best known for his work on ideal numbers, a concept he introduced to address the shortcomings of unique factorization in certain algebraic number fields. This work laid the groundwork for the development of algebraic number theory and influenced later mathematicians such as Richard Dedekind and David Hilbert.
Ideal Numbers and Algebraic Number Theory
Kummer's introduction of ideal numbers was a groundbreaking advancement in the study of algebraic number fields. He was motivated by the failure of unique factorization in the ring of integers of certain number fields, such as the cyclotomic fields. By introducing ideal numbers, Kummer was able to restore the property of unique factorization, which is fundamental to number theory. His work on ideal numbers was later refined and expanded by Dedekind, who introduced the concept of ideals in ring theory.
Fermat's Last Theorem
Kummer made significant progress in the study of Fermat's Last Theorem, a famous problem in number theory. He proved the theorem for a large class of prime exponents, known as regular primes, by using his theory of ideal numbers. Although Kummer did not solve the theorem in its entirety, his work provided crucial insights and techniques that were used by later mathematicians, including Andrew Wiles, who eventually proved the theorem in 1994.
Hypergeometric Functions
In addition to his work in number theory, Kummer made important contributions to the theory of hypergeometric functions. He developed the Kummer confluent hypergeometric function, a special function that arises in the solution of certain differential equations. This work has applications in various fields of mathematics and physics, including the study of wave propagation and quantum mechanics.
Later Life and Legacy
In 1855, Kummer accepted a position at the University of Berlin, where he succeeded Peter Gustav Lejeune Dirichlet as a professor of mathematics. He remained at Berlin for the rest of his career, mentoring many prominent mathematicians, including Leopold Kronecker and Hermann Schwarz. Kummer's influence extended beyond his immediate students, as his work laid the foundation for many developments in algebra and number theory.
Kummer retired from his academic position in 1883 and passed away on May 14, 1893, in Berlin, Germany. His contributions to mathematics have had a lasting impact, particularly in the fields of algebraic number theory and the study of special functions. Kummer's work continues to be studied and built upon by mathematicians today.