Table of Contents
What is the greatest contribution of Kummer in the field of mathematics?
introduction of ideal numbers
On January 29, 1810, German mathematician Ernst Eduard Kummer was born. One of his major contributions is the introduction of ideal numbers, which are defined as a special subgroup of a ring, extended the fundamental theorem of arithmetic to complex number fields.
What exactly is number theory?
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by “number theory”.
What is the modern theory of numbers?
Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, algebraic number theory, analytic number theory, geometric number theory, and probabilistic number theory. These categories reflect the methods used to address problems concerning the integers.
What is the importance of number theory?
Description: The number theory helps discover interesting relationships between different sorts of numbers and to prove that these are true . Number Theory is partly experimental and partly theoretical. Experimental part leads to questions and suggests ways to answer them.
What are the goals of number theory?
The main goal of number theory is to discover interesting and unexpected rela- tionships between different sorts of numbers and to prove that these relationships are true.
What are the key ideas in number theory?
Primes and prime factorization are especially important concepts in number theory. In number theory, a partition is a way of writing a whole number as a sum of positive integers in which the order of the addends is not significant. A perfect number is a positive integer that equals the sum of its divisors.
What is Kummer’s contribution to mathematics?
Kummer also extended the work of Carl Friedrich Gauss on the hypergeometric series, adding developments that are useful in the theory of differential equations. Like Gauss, the German mathematician Ernst Eduard Kummer sought to generalize the law of quadratic reciprocity to deal with questions about third, fourth, and higher powers of numbers.
What did Kummer and Dirichlet discover about ideal numbers?
In 1843 Kummer showed Dirichlet an attempted proof of Fermat’s last theorem, which states that the formula xn + yn = zn, where n is an integer greater than 2, has no solution for positive integral values of x, y, and z. Dirichlet found an error, and Kummer continued his search and developed the concept of ideal numbers.
Who developed the concept of ideal numbers?
But, Dirichlet found an error, and Kummer continued his search and developed the concept of so-called ideal numbers. In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field. An ideal is a special subset of an algebraic ring.
What is so special about the Berlin lectures of Karl Kummer?
Kummer’s Berlin lectures, always carefully prepared, covered analytic geometry, mechanics, the theory of surfaces, and number theory. The clarity and vividness of his presentations brought him great numbers of students – as many as 250 were counted at his lectures.