Kummer's theorem
Introduction
Kummer's theorem, named after the German mathematician Eduard Kummer, is a significant result in the field of number theory. It provides a criterion for the regularity of prime numbers in terms of the divisibility of certain class numbers. This theorem is particularly important in the study of Fermat's Last Theorem and the theory of ideals in algebraic number theory.
Background
Eduard Kummer was a 19th-century mathematician who made significant contributions to number theory, particularly in the study of cyclotomic fields. His work on Fermat's Last Theorem led to the development of the concept of ideals, which are fundamental to modern algebraic number theory.
Kummer's theorem is a result of his work on cyclotomic fields and their class numbers. The theorem provides a criterion for the regularity of a prime number, which is a key concept in the proof of Fermat's Last Theorem.
Statement of the Theorem
Kummer's theorem states that a prime number p is regular if and only if it does not divide the class number of the p-th cyclotomic field.
In more formal terms, let Q(ζ) be the p-th cyclotomic field, where ζ is a primitive p-th root of unity. The class number of Q(ζ) is denoted by h(ζ). Then a prime number p is regular if and only if p does not divide h(ζ).
Proof of the Theorem
The proof of Kummer's theorem involves several steps and makes use of various concepts from algebraic number theory, including the theory of ideals and the properties of cyclotomic fields.
The first step in the proof is to show that the class number h(ζ) is divisible by p if and only if there exists a non-trivial ideal in the ring of integers of Q(ζ) that is stable under the action of the Galois group.
The next step is to show that such an ideal exists if and only if there is a non-trivial solution to a certain system of congruences. This involves the use of Kummer's theory of congruences and the properties of Bernoulli numbers.
Finally, it is shown that a non-trivial solution to this system of congruences exists if and only if p divides the numerator of a certain Bernoulli number. This completes the proof of the theorem.
Applications
Kummer's theorem has several important applications in number theory. One of the most significant is its use in the proof of Fermat's Last Theorem for regular primes.
The theorem also plays a crucial role in the study of cyclotomic fields and their class numbers. It provides a criterion for the regularity of a prime number, which is a key concept in the theory of ideals in algebraic number theory.
In addition, Kummer's theorem is used in the study of elliptic curves and their L-functions, as well as in the theory of modular forms.