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Ernst Kummer | |
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Ernst Eduard Kummer | |

Born | Sorau, Prussia | 29 January 1810

Died | 14 May 1893 83) Berlin, Brandenburg, Germany | (aged

Residence | Germany |

Nationality | Prussian |

Alma mater | University of Halle (Ph.D., 1831) |

Known for | Bessel functions, Kummer theory, Kummer surface, and other contributions |

Scientific career | |

Fields | Mathematician |

Institutions | University of Berlin University of Breslau Gewerbeinstitut Lomonosov University |

Thesis | De cosinuum et sinuum potestatibus secundum cosinus et sinus arcuum multiplicium evolvendis (1831/1832) |

Doctoral advisor | Heinrich Scherk |

Doctoral students | Gotthold Eisenstein Georg Frobenius Lazarus Fuchs Wilhelm Killing Adolf Kneser Franz Mertens Hermann Schwarz Georg Cantor Hans Carl Friedrich von Mangoldt Adolf Piltz Friedrich Prym |

**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.

**Germany**, officially the **Federal Republic of Germany**, is a country in Central and Western Europe, lying between the Baltic and North Seas to the north, and the Alps to the south. It borders Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, France to the southwest, and Luxembourg, Belgium and the Netherlands to the west.

A **mathematician** is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

**Applied mathematics** is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.

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 eventually published a year later.

The **Province of Brandenburg** was a province of Prussia from 1815 to 1945. Brandenburg was established in 1815 from the Kingdom of Prussia's core territory, comprised the bulk of the historic Margraviate of Brandenburg and the Lower Lusatia region, and became part of the German Empire in 1871. From 1918, Brandenburg was a province of the Free State of Prussia until it was dissolved in 1945 after World War II, and replaced with reduced territory as the State of Brandenburg in East Germany, which was later dissolved in 1952. Following the reunification of Germany in 1990, Brandenburg was re-established as a federal state of Germany, becoming one of the new states.

**Prussia** was a historically prominent German state that originated in 1525 with a duchy centred on the region of Prussia on the southeast coast of the Baltic Sea. It was de facto dissolved by an emergency decree transferring powers of the Prussian government to German Chancellor Franz von Papen in 1932 and de jure by an Allied decree in 1947. For centuries, the House of *Hohenzollern* ruled Prussia, successfully expanding its size by way of an unusually well-organised and effective army. Prussia, with its capital in *Königsberg* and from 1701 in Berlin, decisively shaped the history of Germany.

Kummer was married in 1840 to Ottilie Mendelssohn, daughter of Nathan Mendelssohn and Henriette Itzig. Ottilie was a cousin of Felix Mendelssohn and his sister Rebecca Mendelssohn Bartholdy, the wife of the mathematician Peter Gustav Lejeune Dirichlet. His second wife (whom he married soon after the death of Ottilie in 1848), Bertha Cauer, was a maternal cousin of Ottilie. Overall, he had 13 children. His daughter Marie married the mathematician Hermann Schwarz. Kummer retired from teaching and from mathematics in 1890 and died three years later in Berlin.

**Jakob Ludwig Felix Mendelssohn Bartholdy**, born and widely known as **Felix Mendelssohn**, was a German composer, pianist, organist and conductor of the early Romantic period. Mendelssohn's compositions include symphonies, concertos, piano music and chamber music. His best-known works include his Overture and incidental music for *A Midsummer Night's Dream*, the *Italian Symphony*, the *Scottish Symphony*, the oratorio *Elijah*, the overture *The Hebrides*, his mature Violin Concerto, and his String Octet. The melody for the Christmas carol "Hark! The Herald Angels Sing" is also his. Mendelssohn's *Songs Without Words* are his most famous solo piano compositions.

**Johann Peter Gustav Lejeune Dirichlet** was a German mathematician who made deep contributions to number theory, and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function.

**Karl Hermann Amandus Schwarz** was a German mathematician, known for his work in complex analysis.

Kummer made several contributions to mathematics in different areas; he codified some of the relations between different hypergeometric series, known as contiguity relations. The Kummer surface results from taking the quotient of a two-dimensional abelian variety by the cyclic group {1, −1} (an early orbifold: it has 16 singular points, and its geometry was intensively studied in the nineteenth century).

In algebraic geometry, a **Kummer quartic surface**, first studied by Kummer (1864), is an irreducible nodal surface of degree 4 in with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variety of a smooth hyperelliptic curve of genus 2; i.e. a quotient of the Jacobian by the Kummer involution *x* ↦ −*x*. The Kummer involution has 16 fixed points: the 16 2-torsion point of the Jacobian, and they are the 16 singular points of the quartic surface. Resolving the 16 double points of the quotient of a torus by the Kummer involution gives a K3 surface with 16 disjoint rational curves; these K3 surfaces are also sometimes called Kummer surfaces.

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an **abelian variety** is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory.

In the mathematical disciplines of topology, geometry, and geometric group theory, an **orbifold** is a generalization of a manifold. It is a topological space with an orbifold structure.

Kummer also proved Fermat's last theorem for a considerable class of prime exponents (see regular prime, ideal class group). His methods were closer, perhaps, to p-adic ones than to ideal theory as understood later, though the term 'ideal' was invented by Kummer. He studied what were later called Kummer extensions of fields: that is, extensions generated by adjoining an *n*th root to a field already containing a primitive *n*th root of unity. This is a significant extension of the theory of quadratic extensions, and the genus theory of quadratic forms (linked to the 2-torsion of the class group). As such, it is still foundational for class field theory.

In number theory, a **regular prime** is a 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 of either class numbers or of Bernoulli numbers.

In number theory, the **ideal class group** of an algebraic number field *K* is the quotient group *J _{K}*/

In mathematics, the **p-adic number system** for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, p-adic numbers are considered to be close when their difference is divisible by a high power of p: the higher the power, the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.

- Kummer, Ernst Eduard (1975), Weil, André, ed.,
*Collected papers. Volume 1: Contributions to Number Theory*, Berlin, New York: Springer-Verlag, ISBN 978-0-387-06835-0, MR 0465760^{ [1] } - Kummer, Ernst Eduard (1975), Weil, André, ed.,
*Collected papers. Volume II: Function theory, geometry and miscellaneous*, Berlin, New York: Springer-Verlag, ISBN 978-3-540-06836-5, MR 0465761^{ [1] }

**André Weil** was an influential French mathematician of the 20th century, known for his foundational work in number theory and algebraic geometry. He was a founding member and the *de facto* early leader of the mathematical Bourbaki group. The philosopher Simone Weil was his sister.

The **International Standard Book Number** (**ISBN**) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

**Algebraic number theory** is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

In mathematics, **class field theory** is a major branch of algebraic number theory that studies abelian extensions of local fields and "global fields" such as number fields and function fields of curves over finite fields in terms of abelian topological groups associated to the fields. It also studies various arithmetic properties of such abelian extensions. Class field theory includes global class field theory and local class field theory.

**Commutative algebra** is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers ; and *p*-adic integers.

In mathematics, a **reciprocity law** is a generalization of the law of quadratic reciprocity.

**Leopold Kronecker** was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by Weber (1893) as having said, "* Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk*". Kronecker was a student and lifelong friend of Ernst Kummer.

**Oscar Zariski** was a Russian-born American mathematician and one of the most influential algebraic geometers of the 20th century.

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. This means that there is an element of the ring of integers of the Hilbert class field, which is an ideal number, such that the original nonprincipal ideal is equal to the collection of all multiples of this ideal number by elements of this ring of integers that lie in the original field's ring of integers.

In abstract algebra and number theory, **Kummer theory** provides a description of certain types of field extensions involving the adjunction of *n*th roots of elements of the base field. The theory 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 nature of the field – apart from its characteristic, which should not divide the integer *n* – and therefore belong to abstract algebra. The theory of cyclic extensions of the field *K* when the characteristic of *K* does divide *n* is called Artin–Schreier theory.

The **Artin reciprocity law**, which was established by Emil Artin in a series of papers, is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.

**Ernst Witt** was a German mathematician, one of the leading algebraists of his time.

In algebraic number theory, a **reflection theorem** or **Spiegelungssatz** is one of a collection of theorems linking the sizes of different ideal class groups, or the sizes of different isotypic components of a class group. The original example is due to Ernst Eduard Kummer, who showed that the class number of the cyclotomic field , with *p* a prime number, will be divisible by *p* if the class number of the maximal real subfield is. Another example is due to Scholz. A simplified version of his theorem states that if 3 divides the class number of a real quadratic field , then 3 also divides the class number of the imaginary quadratic field .

In mathematics, the **main conjecture of Iwasawa theory** is a deep relationship between *p*-adic *L*-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Mazur and Wiles (1984). The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture. There are several generalizations of the main conjecture, to totally real fields, CM fields, elliptic curves, and so on.

In mathematics, an **algebraic number field***F* is a finite degree field extension of the field of rational numbers **Q**. Thus *F* is a field that contains **Q** and has finite dimension when considered as a vector space over **Q**.

In number theory, a **cyclotomic field** is a number field obtained by adjoining a complex primitive root of unity to **Q**, the field of rational numbers. The n-th cyclotomic field **Q**(ζ_{n}) is obtained by adjoining a primitive n-th root of unity ζ_{n} to the rational numbers.

**Peter Roquette** is a German mathematician working in algebraic geometry, algebra, and number theory.

- 1 2 Mazur, Barry (1977). "Review:
*Kummer, Collected Papers*".*Bull. Amer. Math. Soc*.**83**(5): 976–988. doi:10.1090/s0002-9904-1977-14343-7.

- Eric Temple Bell,
*Men of Mathematics*, Simon and Schuster, New York: 1986. - R. W. H. T. Hudson,
*Kummer's Quartic Surface*, Cambridge, [1905] rept. 1990. - "Ernst Kummer," in
*Dictionary of Scientific Biography*, ed. C. Gillispie, NY: Scribners 1970–90.

- O'Connor, John J.; Robertson, Edmund F., "Ernst Kummer",
*MacTutor History of Mathematics archive*, University of St Andrews . - Works by or about Ernst Kummer at Internet Archive
- Biography of Ernst Kummer
- Ernst Kummer at the Mathematics Genealogy Project

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