Concept
Regular prime
Related concepts (8)
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions. The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica. Fermat added that he had a proof that was too large to fit in the margin.
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem. It was in the process of his deep investigations of the arithmetic of these fields (for prime n) – and more precisely, because of the failure of unique factorization in their rings of integers – that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences.
Ideal number
In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the ring of integers of an algebraic number field is principal if it consists of multiples of a single element of the ring, and nonprincipal otherwise. By the principal ideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field.
Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker. Kummer was born in Sorau, Brandenburg (then part of Prussia). He was awarded a PhD from the University of Halle in 1831 for writing a prize-winning mathematical essay (De cosinuum et sinuum potestatibus secundum cosinus et sinus arcuum multiplicium evolvendis), which was published a year later.
Ring of integers
In mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: . This ring is often denoted by or . Since any integer belongs to and is an integral element of , the ring is always a subring of . The ring of integers is the simplest possible ring of integers. Namely, where is the field of rational numbers. And indeed, in algebraic number theory the elements of are often called the "rational integers" because of this.
Iwasawa theory
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (岩澤 健吉), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently (early 1990s), Ralph Greenberg has proposed an Iwasawa theory for motives.
Ideal class group
In number theory, the ideal class group (or class group) of an algebraic number field K is the quotient group JK/PK where JK is the group of fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of K. The order of the group, which is finite, is called the class number of K. The theory extends to Dedekind domains and their field of fractions, for which the multiplicative properties are intimately tied to the structure of the class group.
Prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4.