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Theorem on powers of primes dividing binomial coefficients
In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other
Kummer's_theorem
Theorem in algebraic number theory
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
Dedekind–Kummer_theorem
Function defined by a hypergeometric series
z = 1 and then using Gauss's theorem to evaluate the result. A typical example is Kummer's theorem, named for Ernst Kummer: 2 F 1 ( a , b ; 1 + a − b ;
Hypergeometric_function
Result on the class group of certain number fields, strengthening Ernst Kummer's theorem
the Herbrand–Ribet theorem is a result on the class group of certain number fields. It is a strengthening of Ernst Kummer's theorem to the effect that
Herbrand–Ribet_theorem
German mathematician (1810–1893)
MR 0465761 25628 Kummer – asteroid named after Ernst Kummer Kummer configuration Kummer's congruence Kummer series Kummer theory Kummer's theorem, on prime-power
Ernst_Kummer
17th-century conjecture proved by Andrew Wiles in 1994
Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Building on Kummer's work
Fermat's_Last_Theorem
Number theory theorem
In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient ( m n ) {\displaystyle {\tbinom {m}{n}}} by a prime
Lucas's_theorem
Rational number sequence
class groups of cyclotomic fields by a theorem of Kummer and its strengthening in the Herbrand-Ribet theorem, and to class numbers of real quadratic
Bernoulli_number
Result in number theory showing congruences involving Bernoulli numbers
integers. Von Staudt–Clausen theorem, another congruence involving Bernoulli numbers Bernoulli number § The Kummer theorems Kummer, Ernst Eduard (1851), "Über
Kummer's_congruence
Theorems that help decompose a finite group based on prime factors of its order
specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow
Sylow_theorems
Criterion for the convergence of a series
version of Kummer's test was established by Tong. See also for further discussions and new proofs. The provided modification of Kummer's theorem characterizes
Ratio_test
theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set
List_of_theorems
Generalization of the binomial theorem to other polynomials
multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from
Multinomial_theorem
Result due to Kummer on cyclic extensions of fields that leads to Kummer theory
Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory.
Hilbert's_Theorem_90
Condition under which an odd prime is a sum of two squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as: p = x 2 + y 2 , {\displaystyle p=x^{2}+y^{2}
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Integers have unique prime factorizations
mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
usually small. Kummer showed that if a prime p {\displaystyle p} does not divide the class number h {\displaystyle h} , then Fermat's Last Theorem holds for
Kummer–Vandiver_conjecture
Product of numbers from 1 to n
formula to the product formula for binomial coefficients produces Kummer's theorem, a similar result on the exponent of each prime in the factorization
Factorial
and only if they have the same multiplicity. Dedekind–Kummer theorem Swanson & Huneke 2006, Theorem 11.3.1 Eisenbud, David, Commutative Algebra with a View
Integral_closure_of_an_ideal
Product of prime numbers, plus one
connection with Euclid's theorem that there are infinitely many prime numbers. A Euclid number of the second kind (also called Kummer number) is an integer
Euclid_number
Correspondence between subfields and subgroups
In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to
Fundamental theorem of Galois theory
Fundamental_theorem_of_Galois_theory
Ideal in a ring which has properties similar to prime elements
countably generated is prime. Radical ideal Maximal ideal Dedekind–Kummer theorem Residue field Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra
Prime_ideal
Number theory expression
\end{aligned}}} Legendre's formula can be used to prove Kummer's theorem. As one special case, it can be used to prove that if n is a positive
Legendre's_formula
Topics referred to by the same term
{R} ^{3}} Hilbert's Theorem 90, an important result on cyclic extensions of fields that leads to Kummer theory Hilbert's basis theorem, in commutative algebra
Hilbert's_theorem
Digit transferred from one column to another
abandoned this experiment, though it remains widely used.[citation needed] Kummer's theorem states that the number of carries involved in adding two numbers in
Carry_(arithmetic)
Identity obeyed by many special functions related to the gamma function
In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit
Multiplication_theorem
Gives information about the Galois module structure of class groups of cyclotomic fields
In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class
Stickelberger's_theorem
Type of prime number
special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility
Regular_prime
Number divisible only by 1 and itself
modulo integer prime numbers. Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected
Prime_number
Field (mathematics) generated by the square root of an integer
|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 to
Quadratic_field
Submodule of fractions in abstract algebra
divisorial ideals is called a Mori domain. Divisorial sheaf Dedekind–Kummer theorem Childress, Nancy (2009). Class field theory. New York: Springer.
Fractional_ideal
Theory in abstract algebra
was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's Last Theorem. The main statements do not depend on the
Kummer_theory
Determines the fractional part of Bernoulli numbers
In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by Karl von Staudt (1840)
Von_Staudt–Clausen_theorem
Theorem in algebraic number theory relating p-adic L-functions and ideal class groups
primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Barry Mazur and Andrew Wiles. The Herbrand–Ribet theorem and the Gras conjecture
Main conjecture of Iwasawa theory
Main_conjecture_of_Iwasawa_theory
Mathematical theorem
established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory
Artin_reciprocity
Gives conditions for the solvability of quadratic equations modulo prime numbers
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations
Quadratic_reciprocity
Solution of a confluent hypergeometric equation
hypergeometric functions: Kummer's (confluent hypergeometric) function M(a, b, z), introduced by Kummer (1837), is a solution to Kummer's differential equation
Confluent hypergeometric function
Confluent_hypergeometric_function
Field extension of the rational numbers by a primitive root of unity
factorization is (Theorem 11.1) 1 through 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 54, 60, 66, 70, 84, 90. Kummer found a way
Cyclotomic_field
German mathematician (1823–1891)
theory he formulated the Kronecker–Weber theorem, without however offering a definitive proof (the theorem was proved completely much later by David
Leopold_Kronecker
Study of objects of arithmetic interest over infinite towers of number fields
{\displaystyle K} had already been identified by Kummer as the main obstruction to the direct proof of Fermat's Last Theorem. From this beginning in the 1950s, a
Iwasawa_theory
One of several theorems linking the sizes of different ideal class groups
theory, a reflection theorem or Spiegelungssatz (German for reflection theorem – see Spiegel and Satz) is one of a collection of theorems linking the sizes
Reflection_theorem
Algebraic integer which represents an ideal in a ring of integers
interest in Fermat's Last Theorem; there is even a story often told that Kummer, like Lamé, believed he had proven Fermat's Last Theorem until Lejeune Dirichlet
Ideal_number
Natural number
numbers (p-adic analysis) Arithmetic Modular arithmetic Chinese remainder theorem Arithmetic functions Advanced concepts Quadratic forms Modular forms L-functions
1
Mathematician (1845–1918)
more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal
Georg_Cantor
German mathematician (1843–1921)
Among other things, Schwarz improved the proof of the Riemann mapping theorem, developed a special case of the Cauchy–Schwarz inequality, and gave a
Hermann_Schwarz
German mathematician (1849–1917)
Padé approximants), and gave the first full proof for the Cayley–Hamilton theorem. He also lent his name to certain differential-geometric objects in modern
Ferdinand_Georg_Frobenius
Branch of Galois theory in mathematics
using additive counterparts of the methods involved in Kummer theory, replacing Hilbert's theorem 90 by the Galois cohomology of the additive group. These
Artin–Schreier_theory
Topics referred to by the same term
larger variability of small samples is neglected. Law of large numbers, a theorem that describes results approaching their average probabilities as they
Law_of_small_numbers
German mathematician (1805–1859)
mathematician. In number theory, he proved special cases of Fermat's Last Theorem and created analytic number theory. In analysis, he advanced the theory
Peter Gustav Lejeune Dirichlet
Peter_Gustav_Lejeune_Dirichlet
Branch of number theory
Ideals generalize Ernst Eduard Kummer's ideal numbers, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem. David Hilbert unified the
Algebraic_number_theory
German mathematician
Fyodorov–Schoenflies–Bieberbach theorem Jordan–Schoenflies theorem Schoenflies notation Schoenflies displacement Heine–Borel theorem Geometrical crystallography
Arthur_Moritz_Schoenflies
Natural number
kind, also called a Kummer number. One standard proof of Euclid's theorem that there are infinitely many primes uses the Kummer numbers, by observing
209_(number)
Branch of mathematics
compositions of analytic functions Montel's theorem Poincaré metric Schwarz lemma Riemann mapping theorem Carathéodory's theorem (conformal mapping) Böttcher's equation
Complex_dynamics
Field of mathematics which studies incidence structures
about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure. Such
Incidence_geometry
Study of computable functions and Turing degrees
reducibilities and other related notions. One of the major results was Kummer's cardinality theorem, which states that a set A is computable if and only if there
Computability_theory
Mathematics of varieties with integer coordinates
geometry. Four theorems of fundamental importance in Diophantine geometry are: Mordell–Weil theorem Roth's theorem Siegel's theorem Faltings' theorem Another
Diophantine_geometry
Branch of mathematics
example, Sylow's theorem was reproven by Frobenius in 1887 directly from the laws of a finite group, although Frobenius remarked that the theorem followed from
Abstract_algebra
Prime pair of the form (p, 2p+1)
work was the most progress achieved on Fermat's last theorem at that time. Later work by Kummer and others always divided the problem into first and second
Safe and Sophie Germain primes
Safe_and_Sophie_Germain_primes
Finite extension of the rationals
field Dirichlet's unit theorem, S-unit Kummer extension Minkowski's theorem, Geometry of numbers Chebotarev's density theorem Ray class group Decomposition
Algebraic_number_field
Algebraic variety of dimension two
quadratic form. This theorem is proven using the Nakai criterion and the Riemann-Roch theorem for surfaces. The Hodge index theorem is used in Deligne's
Algebraic_surface
Abelian group
some long exact sequences from homological algebra and the Kummer map. There are many theorems in the literature about the structure of the Mordell–Weil
Mordell–Weil_group
On finite sums of products of three binomial coefficients, and a hypergeometric sum
> 0. As c tends to −∞ it reduces to Kummer's formula for the hypergeometric function 2F1 at −1. Dixon's theorem can be deduced from the evaluation of
Dixon's_identity
Type of smooth complex surface of kodaira dimension 0
s'agit des variétés kählériennes dites K3, ainsi nommées en l'honneur de Kummer, Kähler, Kodaira et de la belle montagne K2 au Cachemire. In the second
K3_surface
Sum in algebraic number theory
Gauss sum Jacobi sum Kummer sum Kloosterman sum Gaussian period Hasse–Davenport relation Chowla–Mordell theorem Stickelberger's theorem B. H. Gross and N
Gauss_sum
Algebra with unique prime factorization
is a UFD is now known). At the same time, Kummer developed powerful new methods to prove Fermat's Last Theorem at least for a large class of prime exponents
Dedekind_domain
Branch of algebraic number theory concerned with abelian extensions
subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem). One of the major results is: given a number field F, and writing K for
Class_field_theory
German mathematician (1859–1937)
by Jensen. Hölder is also noted for many other theorems including the Jordan–Hölder theorem, the theorem stating that every linearly ordered group that
Otto_Hölder
extend the Kronecker–Weber theorem on Abelian extensions of Q {\displaystyle \mathbb {Q} } to any base number field. Kummer–Vandiver conjecture: primes
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
In number theory, measure of non-unique factorization
Last Theorem by factorisation using the roots of unity was for a very good reason: a failure of unique factorization – i.e., the fundamental theorem of
Ideal_class_group
Number
Kaplan 2000, p. 68–75. Roy, Rahul (January 2003). "Babylonian Pythagoras' Theorem, the Early History of Zero and a Polemic on the Study of the History of
0
as of September 2022[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic
List_of_conjectures
Algebraic structure with addition, multiplication, and division
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
Field_(mathematics)
German mathematician (1854–1925)
number theorem. Mangoldt completed his Doctorate of Philosophy (Ph.D) in 1878 at the University of Berlin, where his supervisors were Ernst Kummer and Karl
Hans Carl Friedrich von Mangoldt
Hans_Carl_Friedrich_von_Mangoldt
Galois extension whose Galois group is abelian
semi-direct product. Kummer theory gives a complete description of the abelian extension case, and the Kronecker–Weber theorem tells us that if K is
Abelian_extension
Branch of mathematics that studies algebraic structures
basis theorem Hopkins–Levitzki theorem Krull's principal ideal theorem Levitzky's theorem Galois theory Abel–Ruffini theorem Wedderburn–Artin theorem Jacobson
List of abstract algebra topics
List_of_abstract_algebra_topics
Branch of pure mathematics
understand but are very difficult to solve. Examples of this are Fermat's Last Theorem, which was proved 358 years after the original formulation, and Goldbach's
Number_theory
Family of power series in mathematics
methodology of proving these identities is the Egorychev method. Saalschütz's theorem (Saalschütz 1890) is 3 F 2 ( a , b , − n ; c , 1 + a + b − c − n ; 1 )
Generalized hypergeometric function
Generalized_hypergeometric_function
1798 textbook by Carl Friedrich Gauss
many theorems with numerical examples. The Disquisitiones was the starting point for other 19th-century European mathematicians, including Ernst Kummer, Peter
Disquisitiones_Arithmeticae
German physician and mathematician
paper by Ernst Kummer and abandoned the attempt. The paper had detected a flaw in Augustin Cauchy's attempted proof of Fermat's Last Theorem, and afterwards
Paul_Wolfskehl
Branch of algebra that studies commutative rings
Lasker–Noether theorem, given here, may be seen as a certain generalization of the fundamental theorem of arithmetic: Lasker-Noether Theorem—Let R be a commutative
Commutative_algebra
corresponding function field extension) is cyclic. The fundamental theorem of Kummer theory implies [citation needed] that a superelliptic curve of degree
Superelliptic_curve
Japanese mathematician (1915–1997)
the theorem that they form a single diffeomorphism class. Again, this work has proved foundational. (The K3 surfaces were named after Ernst Kummer, Erich
Kunihiko_Kodaira
Bernhard Riemann. Koizumi's theorem states the third power of an ample line bundle is normally generated. The Mumford–Kempf theorem states that the fourth
Equations defining abelian varieties
Equations_defining_abelian_varieties
Types of special mathematical functions
and locally the sum converges uniformly for all complex s and x. By a theorem of Weierstrass, the limiting function, sometimes denoted as γ ∗ {\displaystyle
Incomplete_gamma_function
characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In
Normal_basis
Mathematical group that can be generated as the set of powers of a single element
Golubitsky 2010, pp. 47–48). (Cox 2012, p. 294, Theorem 11.1.7). (Cox 2012, p. 295, Corollary 11.1.8 and Theorem 11.1.9). (Aluffi 2009, pp. 82–84, 6.4 Example:
Cyclic_group
Branch of mathematics that studies the properties of groups
to class groups and regular primes, which feature in Kummer's treatment of Fermat's Last Theorem. Analysis on Lie groups and certain other groups is called
Group_theory
Mathematical law, a generalization of quadratic reciprocity
{q-1}{2}}}} is 1 {\displaystyle 1} or − 1 {\displaystyle -1} . By the factor theorem and the behavior of degrees in factorizations the solubility of such quadratic
Reciprocity_law
Type of geometry
(complex) projective geometry, and some theorems about circles can be considered as special cases of these general theorems. During the early 19th century the
Projective_geometry
theory and Stickelberger's theorem as a theory of ideal class groups as Galois modules and p-adic L-functions (with roots in Kummer congruence on Bernoulli
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Formula in number theory
unity contained in K. DK is the discriminant of the extension K/Q. Then: Theorem (Class Number Formula). ζK(s) converges absolutely for Re(s) > 1 and extends
Class_number_formula
Problem about mathematical number fields
455 Hilbert's twelfth problem is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field
Hilbert's_twelfth_problem
Branch of algebraic geometry
modularity theorem) relating elliptic curves to modular forms. This connection would ultimately lead to the first proof of Fermat's Last Theorem in number
Arithmetic_geometry
Theory in number theory
Masatoshi Gündüz Ikeda, Kenkichi Iwasawa, and Kôji Uchida (Neukirch–Uchida theorem, 1969), prior to conjectures made about hyperbolic curves over number fields
Anabelian_geometry
German mathematician (1862–1930)
for the first proof of the four-vertex theorem that applied in general to non-convex curves. Kneser's theorem on differential equations is named after
Adolf_Kneser
cohomology Hasse norm theorem Herbrand quotient Hilbert class field Kronecker–Weber theorem Local class field theory Takagi existence theorem Tate cohomology
Class_formation
Points and lines with equal incidences
book Geometrie der Lage, in the context of a discussion of Desargues' theorem. Ernst Steinitz wrote his dissertation on the subject in 1894, and they
Configuration_(geometry)
German mathematician and physicist
Schumann. Runge's law Runge's method for Diophantine equations. Runge's theorem for complex analysis Ueber die Krümmung, Torsion und geodätische Krümmung
Carl_Runge
quasielliptic counterexamples to the conclusions of the Kodaira vanishing theorem Exceptional surfaces, surfaces whose Picard number achieve the bound set
List of complex and algebraic surfaces
List_of_complex_and_algebraic_surfaces
French mathematician (1859–1921)
Paris, France) was a French mathematician who worked on Kummer surfaces and the Appell–Humbert theorem and introduced Humbert surfaces. His son was the mathematician
Marie_Georges_Humbert
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