Ernst Kummer | |
---|---|

Born | Ernst Eduard Kummer 29 January 1810 |

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

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.

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.

In 1840, Kummer married 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.

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).

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.

Kummer further conducted research in ballistics and, jointly with William Rowan Hamilton he investigated ray systems.^{ [1] }

- 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^{ [2] } - 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^{ [2] }

- Kummer configuration
- Kummer's congruence
- Kummer series
- Kummer theory
- Kummer's theorem, on prime-power divisors of binomial coefficients
- Kummer's function
- Kummer sum
- Kummer variety
- Kummer–Vandiver conjecture
- Kummer's transformation of series
- Ideal number
- Regular prime
- Reflection theorem
- Principalization
- 25628 Kummer – asteroid named after Ernst Kummer

**André Weil** was a French mathematician, 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 writer Sylvie Weil is his daughter.

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

**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.

**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.

In mathematics, **class field theory** is the branch of algebraic number theory concerned with describing the Galois extensions of local and global fields. Hilbert is often credited for the notion of *class field*. But it was already familiar for Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. This theory has its origins in the proof of quadratic reciprocity by Gauss at the end of the 18th century. These ideas were developed over the next century, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin. These conjectures and their proofs constitute the main body of 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 to arbitrary monic irreducible polynomials with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial splits into linear terms when reduced mod . That is, it determines which prime numbers the relation

**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.

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 Kenkichi Iwasawa (1959), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently, Ralph Greenberg has proposed an Iwasawa theory for motives.

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.

In mathematics, the **Kummer–Vandiver conjecture**, or **Vandiver conjecture**, states that a prime *p* does not divide the class number *h _{K}* of the maximal real subfield of the

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.

This is a glossary of **arithmetic and diophantine geometry** in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.

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

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 number theory, a **cyclotomic field** is a number field obtained by adjoining a complex root of unity to **Q**, the field of rational numbers.

In mathematics, the * Zahlbericht* was a report on algebraic number theory by Hilbert.

- ↑ E. E. Kummer:
*Über die Wirkung des Luftwiderstandes auf Körper von verschiedener Gestalt, ins besondere auch auf die Geschosse*, In:*Mathematische Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin*, 1875 - 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.

- Media related to Ernst Kummer at Wikimedia Commons
- 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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