Arnold Scholz (24 December 1904 in

Berlin Berlin ( ; ) is the Capital of Germany, capital and largest city of Germany, by both area and List of cities in Germany by population, population. With 3.7 million inhabitants, it has the List of cities in the European Union by population withi ...

– 1 February 1942 in

Flensburg Flensburg (; Danish language, Danish and ; ; ) is an independent city, independent town in the far north of the Germany, German state of Schleswig-Holstein. After Kiel and Lübeck, it is the third-largest city in Schleswig-Holstein. Flensburg's ...

) was a German mathematician working in

algebraic number theory 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 ob ...

who proved Scholz's reciprocity law and introduced the

Scholz conjecture In mathematics, the Scholz conjecture is a conjecture on the length of certain addition chains. It is sometimes also called the Scholz–Brauer conjecture or the Brauer–Scholz conjecture, after Arnold Scholz, who formulated it in 1937, and Alf ...

Biography

Life

Scholz was the son of Reinhold Scholz, an executive at the

Prussian Prussia (; ; Old Prussian: ''Prūsija'') was a German state centred on the North European Plain that originated from the 1525 secularization of the Prussian part of the State of the Teutonic Order. For centuries, the House of Hohenzoll ...

Military Research Office. He attended the Kaiserin Auguste Gymnasium in

Charlottenburg Charlottenburg () is a Boroughs and localities of Berlin, locality of Berlin within the borough of Charlottenburg-Wilmersdorf. Established as a German town law, town in 1705 and named after Sophia Charlotte of Hanover, Queen consort of Kingdom ...

. From 1923 to 1928, he studied Mathematics, Philosophy and

Musicology Musicology is the academic, research-based study of music, as opposed to musical composition or performance. Musicology research combines and intersects with many fields, including psychology, sociology, acoustics, neurology, natural sciences, ...

at the Universität Berlin. In 1928, Scholz wrote his dissertation, supervised by

Issai Schur Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the Humboldt University of Berlin, University of Berlin. He obtained his doctorate in 1901, became lecturer i ...

and titled ''Über die Bildung algebraischer Zahlkörper mit auflösbarer Galoisscher Gruppe'' ("On the creation of

algebraic number fields In mathematics, an algebraic number field (or simply number field) is an extension field K of the field (mathematics), field of rational numbers such that the field extension K / \mathbb has Degree of a field extension, finite degree (and hence ...

with solvable Galois Groups"). In 1927, Scholz spent a semester in

Vienna Vienna ( ; ; ) is the capital city, capital, List of largest cities in Austria, most populous city, and one of Federal states of Austria, nine federal states of Austria. It is Austria's primate city, with just over two million inhabitants. ...

, where he studied under

Philipp Furtwängler Friederich Pius Philipp Furtwängler (April 21, 1869 – May 19, 1940) was a German number theorist. Biography Furtwängler wrote an 1896 doctoral dissertation at the University of Göttingen on cubic forms (''Zur Theorie der in Linearfaktoren z ...

. After his promotion, he became a lecture assistant in Berlin in 1928, a

Privatdozent ''Privatdozent'' (for men) or ''Privatdozentin'' (for women), abbreviated PD, P.D. or Priv.-Doz., is an academic title conferred at some European universities, especially in German-speaking countries, to someone who holds certain formal qualifi ...

Freiburg im Breisgau Freiburg im Breisgau or simply Freiburg is the List of cities in Baden-Württemberg by population, fourth-largest city of the German state of Baden-Württemberg after Stuttgart, Mannheim and Karlsruhe. Its built-up area has a population of abou ...

in 1930, and a lecturer at the

University of Kiel Kiel University, officially the Christian Albrecht University of Kiel, (, abbreviated CAU, known informally as Christiana Albertina) is a public research university in the city of Kiel, Germany. It was founded in 1665 as the ''Academia Holsator ...

from 1935-1940. In 1940, he was

conscripted Conscription, also known as the draft in the United States and Israel, is the practice in which the compulsory enlistment in a national service, mainly a military service, is enforced by law. Conscription dates back to antiquity and it contin ...

and became a lecturer at the

Mürwik Naval School The Naval Academy Mürwik () is the main training establishment for all German Navy officers and in 1910 replaced the German Imperial Naval Academy in Kiel (which is now the seat of government or ''Landeshaus'' of Schleswig-Holstein). It is lo ...

- Mürwik. Throughout his life, he exchanged many letters with

Helmut Hasse Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ...

, and in the 1930s he worked with

Olga Taussky-Todd Olga Taussky-Todd (August 30, 1906 – October 7, 1995) was an Austrian and later Czech Americans, Czech-American mathematician. She published more than 300 research papers on algebraic number theory, integral matrices, and Matrix (mathematics), ...

. In 1942, Scholz died of

Diabetes Diabetes mellitus, commonly known as diabetes, is a group of common endocrine diseases characterized by sustained high blood sugar levels. Diabetes is due to either the pancreas not producing enough of the hormone insulin, or the cells of th ...

Work

Scholz worked in

. He published early works on the

inverse Galois problem In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved. There ...

, where, together with Hans Reichardt, he proved the solvability of the problem for p-groups in the case where ''p'' is an odd prime. After the end of

World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...

, the works of Reichardt and Scholz were carried on by

Igor Shafarevich Igor Rostislavovich Shafarevich (; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. Outside mathematics, he wrote books and articles that criticised social ...

, who showed that the problem is solvable for all

solvable groups In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group (mathematics), group that can be constructed from abelian groups using Group extension, extensions. Equivalently, a solvable group is a ...

. In 1928, Scholz proved the existence of algebraic number fields with arbitrarily large class field towers. In 1930, Scholz participated in the Second Conference on the Epistemology of the Exact Sciences contributing the paper "On the Use of the Term Holism in Axiomatics" to the discussion on the

foundation of mathematics Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particul ...

. Scholz's reciprocity law is named after him, although according to historian of mathematics Franz LemmermeyerLemmermeyer ''Reziprozitätsgesetze'', Springer Verlag, 2000, p. 160, the law was already known to Theodor Schönemann. His ''

Nachlass ''Nachlass'' (, older spelling ''Nachlaß'') is a German language, German word, used in academia to describe the collection of manuscripts, notes, correspondence, and so on left behind when a scholar dies. The word is a compound word, compound in ...

'' included an almost complete manuscript titled ''Spezielle Zahlkörper'' (Special Number Fields), intended for the newest edition of the Encyclopedia of Mathematical Sciences.

Publications

References

External links

* 1942 deaths 1904 births 20th-century German mathematicians Deaths from diabetes in Germany {{Germany-mathematician-stub