In algebraic number theory, the Dedekind–Kummer theorem describes how a prime ideal in a Dedekind domain factors over the domain's integral closure. It...

that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial...

on divisorial ideals is called a Mori domain. Divisorial sheaf Dedekind-Kummer theorem Childress, Nancy (2009). Class field theory. New York: Springer...

Eduard Kummer's ideal numbers, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem. (Thus Dedekind can be said to have been Kummer's most...

principal ideal domains. In 1843 Kummer introduced the concept of ideal number, which was developed further by Dedekind (1876) into the modern theory of...

not countably generated is prime. Radical ideal Maximal ideal Dedekind–Kummer theorem Residue field Dummit, David S.; Foote, Richard M. (2004). Abstract...

(see regular prime). Somewhat later again Richard Dedekind formulated the concept of an ideal, Kummer having worked in a different way. At this point the...

182). Dedekind gave at least two proofs based on the arithmetic of the Gaussian integers. There is an elegant proof using Minkowski's theorem about convex...

field Dirichlet's unit theorem, S-unit Kummer extension Minkowski's theorem, Geometry of numbers Chebotarev's density theorem Ray class group Decomposition...

Richard Dedekind, Ernst Kummer, and David Hilbert to the study of general algebraic number fields and their rings of integers; specifically Kummer invented...

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...

if and only if they have the same multiplicity. Dedekind–Kummer theorem Swanson & Huneke 2006, Theorem 11.3.1 Eisenbud, David, Commutative Algebra with...

century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet and Richard Dedekind. Many of the annotations given by Gauss are in...

),&d>0.\end{cases}}} This is a special case of Theorem 1 above: for a quadratic field K, the Dedekind zeta function is just ζ K ( s ) = ζ ( s ) L ( s...

|p|<M_{k}.} page 72 These decompositions can be found using the Dedekind–Kummer theorem. A classical example of the construction of a quadratic field is...

1893, as well as simple groups. He also completed the Jordan–Hölder theorem. Dedekind and Miller independently characterized Hamiltonian groups and introduced...

mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In analysis, he advanced the theory...

well-ordering principle as a theorem and attempted to prove it. In 1899, he sent Dedekind a proof of the equivalent aleph theorem: the cardinality of every...

by the Abel–Ruffini theorem). In algebraic number theory Kronecker introduced the theory of divisors as an alternative to Dedekind's theory of ideals, which...

modulo integer prime numbers. Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected...

many theorems with numerical examples. The Disquisitiones was the starting point for other 19th-century European mathematicians, including Ernst Kummer, Peter...

objects in geometry such as points at infinity. In 1876, Richard Dedekind replaced Kummer's undefined concept by concrete sets of numbers, sets that he called...

all 0 < m < n. For n being a regular prime, Kummer used cyclotomic fields to prove Fermat's Last Theorem, which asserts the non-existence of rational...

ideal theorem Levitzky's theorem Galois theory Abel–Ruffini theorem Artin-Wedderburn theorem Jacobson density theorem Wedderburn's little theorem Lasker–Noether...

Dedekind defined the concept of the ring of integers of a number field. In this context, he introduced the terms "ideal" (inspired by Ernst Kummer's notion...

as ideal theory, began with Richard Dedekind's work on ideals, itself based on the earlier work of Ernst Kummer and Leopold Kronecker. Later, David Hilbert...

Weierstrass (by his pupil E. Kossak), Eduard Heine, Georg Cantor, and Richard Dedekind was brought about. In 1869, Charles Méray had taken the same point of departure...

the relation with the theory of modules that had just been developed by Dedekind. Three joint papers with Frobenius deal with the theory of elliptic functions...

factorization in some cyclotomic fields led Ernst Kummer to the concept of ideal numbers and, later, Richard Dedekind to ideals. The quadratic integer rings are...

E/F corresponds to an extension k⊂K of global fields. By studying the Dedekind zeta function of K one shows that the degree 1 primes of K have Dirichlet...