Nikolaus Hofreiter
Encyclopedia
Nikolaus Hofreiter was an Austrian mathematician who worked mainly in number theory
.
Hofreiter went to school in Linz and studied from 1923 in Vienna
with Hans Hahn
, Wilhelm Wirtinger
, Emil Müller at the Technische Universität Wien on descriptive geometry, and Philipp Furtwängler
, with whom he obtained his doctorate in 1927 on the reduction theory of quadratic forms (Eine neue Reduktionstheorie für definite quadratische Formen). In 1928 he passed the Lehramtsprüfung examination and completed the probationary year as a teacher in Vienna, but then returned to the university (first as a scientific assistant at the TU Vienna) where in 1929 he was assistant to Furtwängler and then habilitated in 1933. He was even then an excellent teacher, and gave lectures not only in Vienna but also in Graz
.
His dissertation and habilitation thesis dealt with the reduction theory of quadratic forms, which Gauss
, Charles Hermite
and Hermann Minkowski
had worked on previously. Hofreiter treated the case of four variables of a problem of Minkowski (Minkowski had solved the problem for two variables, while Robert Remak
had solved it for three variables) on the product of inhomogeneous linear forms and achieved significant progress. The complete solution was only found 15 years later (and the general case is still unresolved). In 1934, he proved the existence of infinitely many real quadratic number fields without a Euclidean algorithm
. In addition, he dealt with the geometry of numbers
and Diophantine approximation
.
In 1939, he was an associate professor and married the mathematician Margaret Dostalík (born 1912). She was also a student of Furtwängler and did important work on algebraic equations and was working as a meteorologist in Berlin
at the time. During the Second World War, he moved from Vienna and was a little later at the Hermann Goering Aviation Research Institute in Braunschweig
, where his colleagues Wolfgang Gröbner
from Vienna, Bernhard Baule from Graz, Ernst Peschl and Josef Laub were already working. Through his work there, together with Gröbner, he started a table of integrals. The first volume, on indefinite integrals, was published by Notdruck (Braunschweig) in 1944 and by Springer in 1949. In 1950, the second volume containing definite integrals appeared. Both parts were widely available through to the 5th 1973/75 edition. His wife, Margaret, assisted with the calculations, as well as the preparation and review of both volumes.
In addition to their work at the Aviation Research Institute, Gröbner and Hofreiter continued to give lectures and seminars at the Technical University of Braunschweig. After the war he returned to Vienna in 1946 and continued to expand on his number theory work. He also worked on linear optimization and numerical mathematics. In 1954, he became a professor, and in 1963/4 became Dean of the Faculty, and in 1965/6, he was Rector of the Universität Wien. In 1974, he retired.
In 1970, he corresponding member of the Austrian Academy of Sciences. He received the Honorary Cross for Science and Art 1st Class, the Medal of Honor of the City of Vienna in gold, the great silver medals for outstanding contributions to the Republic of Austria, the Komturkreuz of Gregory, the ring of honor of the Austrian Mathematical Society and was honorary senator of the University of Linz.
His doctoral students included Edmund Hlawka
, Peter Gruber and Leopold Schmetterer.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
.
Hofreiter went to school in Linz and studied from 1923 in Vienna
Vienna
Vienna is the capital and largest city of the Republic of Austria and one of the nine states of Austria. Vienna is Austria's primary city, with a population of about 1.723 million , and is by far the largest city in Austria, as well as its cultural, economic, and political centre...
with Hans Hahn
Hans Hahn
Hans Hahn was an Austrian mathematician who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory.-Biography:...
, Wilhelm Wirtinger
Wilhelm Wirtinger
Wilhelm Wirtinger was an Austrian mathematician, working in complex analysis, geometry, algebra, number theory, Lie groups and knot theory.-Biography:...
, Emil Müller at the Technische Universität Wien on descriptive geometry, and Philipp Furtwängler
Philipp Furtwängler
Philipp Furtwängler was a German number theorist.He wrote an 1896 doctoral dissertation at the University of Göttingen on cubic forms , under Felix Klein...
, with whom he obtained his doctorate in 1927 on the reduction theory of quadratic forms (Eine neue Reduktionstheorie für definite quadratische Formen). In 1928 he passed the Lehramtsprüfung examination and completed the probationary year as a teacher in Vienna, but then returned to the university (first as a scientific assistant at the TU Vienna) where in 1929 he was assistant to Furtwängler and then habilitated in 1933. He was even then an excellent teacher, and gave lectures not only in Vienna but also in Graz
Graz
The more recent population figures do not give the whole picture as only people with principal residence status are counted and people with secondary residence status are not. Most of the people with secondary residence status in Graz are students...
.
His dissertation and habilitation thesis dealt with the reduction theory of quadratic forms, which Gauss
Gauss
Gauss may refer to:*Carl Friedrich Gauss, German mathematician and physicist*Gauss , a unit of magnetic flux density or magnetic induction*GAUSS , a software package*Gauss , a crater on the moon...
, Charles Hermite
Charles Hermite
Charles Hermite was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra....
and Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
had worked on previously. Hofreiter treated the case of four variables of a problem of Minkowski (Minkowski had solved the problem for two variables, while Robert Remak
Robert Remak
-External links:*** in the Virtual Laboratory of the Max Planck Institute for the History of Science...
had solved it for three variables) on the product of inhomogeneous linear forms and achieved significant progress. The complete solution was only found 15 years later (and the general case is still unresolved). In 1934, he proved the existence of infinitely many real quadratic number fields without a Euclidean algorithm
Euclidean algorithm
In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
. In addition, he dealt with the geometry of numbers
Geometry of numbers
In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....
and Diophantine approximation
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
.
In 1939, he was an associate professor and married the mathematician Margaret Dostalík (born 1912). She was also a student of Furtwängler and did important work on algebraic equations and was working as a meteorologist in Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
at the time. During the Second World War, he moved from Vienna and was a little later at the Hermann Goering Aviation Research Institute in Braunschweig
Braunschweig
Braunschweig , is a city of 247,400 people, located in the federal-state of Lower Saxony, Germany. It is located north of the Harz mountains at the farthest navigable point of the Oker river, which connects to the North Sea via the rivers Aller and Weser....
, where his colleagues Wolfgang Gröbner
Wolfgang Gröbner
Wolfgang Gröbner was an Austrian mathematician. His name is best known for the Gröbner basis, used for computations in algebraic geometry...
from Vienna, Bernhard Baule from Graz, Ernst Peschl and Josef Laub were already working. Through his work there, together with Gröbner, he started a table of integrals. The first volume, on indefinite integrals, was published by Notdruck (Braunschweig) in 1944 and by Springer in 1949. In 1950, the second volume containing definite integrals appeared. Both parts were widely available through to the 5th 1973/75 edition. His wife, Margaret, assisted with the calculations, as well as the preparation and review of both volumes.
In addition to their work at the Aviation Research Institute, Gröbner and Hofreiter continued to give lectures and seminars at the Technical University of Braunschweig. After the war he returned to Vienna in 1946 and continued to expand on his number theory work. He also worked on linear optimization and numerical mathematics. In 1954, he became a professor, and in 1963/4 became Dean of the Faculty, and in 1965/6, he was Rector of the Universität Wien. In 1974, he retired.
In 1970, he corresponding member of the Austrian Academy of Sciences. He received the Honorary Cross for Science and Art 1st Class, the Medal of Honor of the City of Vienna in gold, the great silver medals for outstanding contributions to the Republic of Austria, the Komturkreuz of Gregory, the ring of honor of the Austrian Mathematical Society and was honorary senator of the University of Linz.
His doctoral students included Edmund Hlawka
Edmund Hlawka
Edmund Hlawka was an Austrian mathematician. He was a leading number theorist. Hlawka did most of his work at the Vienna University of Technology. He was also a visiting professor at Princeton University and the Sorbonne...
, Peter Gruber and Leopold Schmetterer.