Helmut Hasse

# Helmut Hasse

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Helmut Hasse (ˈhasə; 25 August 1898 – 26 December 1979) was a German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

working in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

, known for fundamental contributions to class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...

, the application of p-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

s to local classfield theory
Local classfield theory
In mathematics, local class field theory, introduced by , is the study of abelian extensions of local fields. It is the analogue for local fields of global class field theory....

and diophantine geometry
Diophantine geometry
In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general...

(Hasse principle
Hasse principle
In mathematics, Helmut Hasse's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number...

), and to local zeta functions.

## Life

He was born in Kassel
Kassel
Kassel is a town located on the Fulda River in northern Hesse, Germany. It is the administrative seat of the Kassel Regierungsbezirk and the Kreis of the same name and has approximately 195,000 inhabitants.- History :...

, and died in Ahrensburg
Ahrensburg
Ahrensburg is a town in Schleswig-Holstein, Germany. It is part of the Hamburg Metropolitan Region, situated in Stormarn. Its population around 31,700 . Its outstanding sight is the Renaissance castle dating from 1595....

.

After serving in the navy in World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...

, he studied at the University of Göttingen, and then at Marburg under Kurt Hensel
Kurt Hensel
Kurt Wilhelm Sebastian Hensel was a German mathematician born in Königsberg, Prussia.He was the son of the landowner and entrepreneur Sebastian Hensel, brother of the philosopher Paul Hensel, grandson of the composer Fanny Mendelssohn and the painter Wilhelm Hensel, and a descendant of the...

, writing a dissertation in 1921 containing the Hasse–Minkowski theorem
Hasse–Minkowski theorem
The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent locally at all places, i.e. equivalent over every completion of the field...

, as it is now called, on quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

s over number fields. He then held positions at Kiel
University of Kiel
The University of Kiel is a university in the city of Kiel, Germany. It was founded in 1665 as the Academia Holsatorum Chiloniensis by Christian Albert, Duke of Holstein-Gottorp and has approximately 23,000 students today...

, Halle
University of Halle-Wittenberg
The Martin Luther University of Halle-Wittenberg , also referred to as MLU, is a public, research-oriented university in the cities of Halle and Wittenberg within Saxony-Anhalt, Germany...

and Marburg. He was Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...

's replacement at Göttingen in 1934; politically he was a right-wing nationalist, and applied for membership in the Nazi Party in 1937 but this was denied to him because he had Jewish ancestors. After war work he returned to Göttingen briefly in 1945 but was excluded by the British authorities. After brief appointments in Berlin from 1948 he settled permanently as professor in Hamburg
University of Hamburg
The University of Hamburg is a university in Hamburg, Germany. It was founded on 28 March 1919 by Wilhelm Stern and others. It grew out of the previous Allgemeines Vorlesungswesen and the Kolonialinstitut as well as the Akademisches Gymnasium. There are around 38,000 students as of the start of...

.

He collaborated with many mathematicians: in particular with Emmy Noether
Emmy Noether
Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...

and Richard Brauer
Richard Brauer
Richard Dagobert Brauer was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory...

on simple algebras; and with Harold Davenport
Harold Davenport
Harold Davenport FRS was an English mathematician, known for his extensive work in number theory.-Early life:...

on Gauss sum
Gauss sum
In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typicallyG := G= \sum \chi\cdot \psi...

s (Hasse–Davenport relation
Hasse–Davenport relation
The Hasse–Davenport relations, introduced by , are two related identities for Gauss sums, one called the Hasse–Davenport lifting relation, and the other called the Hasse–Davenport product relation. The Hasse–Davenport lifting relation is an equality in number theory relating Gauss sums over...

s) and with Cahit Arf
Cahit Arf
Cahit Arf was a Turkish mathematician. He is known for the Arf invariant of a quadratic form in characteristic 2 in topology, the Hasse–Arf theorem in ramification theory, Arf semigroups, and Arf rings.-Biography:Cahit Arf was born on 11 October 1910 in Selanik , which was then...

on the Hasse–Arf theorem
Hasse–Arf theorem
In mathematics, specifically in local class field theory, the Hasse–Arf theorem is a result concerning jumps of a filtration of the Galois group of a finite Galois extension...

.

• Hasse diagram
Hasse diagram
In order theory, a branch of mathematics, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction...

• Hasse invariant of an elliptic curve
• Hasse invariant of a quadratic form
Hasse invariant of a quadratic form
In mathematics, the Hasse invariant of a quadratic form Q over a field K takes values in the Brauer group Br.The quadratic form Q may be taken as a diagonal form...

• Artin–Hasse exponential
• Hasse–Weil L-function
• Hasse norm theorem
Hasse norm theorem
In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm....

• Hasse's algorithm
• Hasse's theorem on elliptic curves