Helmut Hasse: Bringing p-adic numbers into the heart of number theory
Helmut Hasse was born on August 25, 1898, in Kassel, Hesse-Nassau, German Empire, as the son of judge Paul Reinhard Hasse. His mother was born in Milwaukee, United States. Hasse became one of the most important mathematicians of the 20th century, with roles encompassing research, dissemination of mathematical knowledge, teaching, and academic journal editing.
From sailor to mathematician
After serving in the navy during World War I, Hasse studied at the University of Göttingen before moving to continue his studies at the University of Marburg under the supervision of Kurt Hensel, a pioneer of p-adic number theory. In 1921, Hasse wrote a thesis that contained what is now known as the Hasse-Minkowski theorem concerning quadratic forms over number fields. After graduation, he held positions at the University of Kiel, the University of Halle, and the University of Marburg, respectively.
Hassel Principle: From Local to Global
Hasse's most famous work is the Hasse principle, also called the "local-global principle." This concept states that certain types of Diophantine equations have rational solutions if and only if they have solutions in every local extension of the rational numbers, both real numbers and p-adic numbers for every prime p. This principle made Hensel's p-adic numbers an indispensable tool in modern number theory and paved the way for the development of the Hasse-Minkowski theorem, which proves that this principle actually applies to quadratic forms over the field of rational numbers. This theorem became a fundamental foundation of Diophantine geometry and remains a basic tool that number theorists use to this day.
Pioneer of field class theory
Aside from Hasse's theorem, he also made fundamental contributions to class field theory, which is an important branch of algebraic number theory that studies the relationship between number fields and groups of ideals. He also successfully proved the Riemann hypothesis for elliptic curves, which is highly significant in arithmetic algebraic geometry. Additionally, Hasse collaborated with many other leading mathematicians on several works, such as the Artin-Hasse exponential with Emil Artin, the Hasse-Weil L-function with André Weil, the Hasse-Arf theorem with Cahit Arf, and the Hasse-Davenport relation with Harold Davenport.
A period of conflict in the Nazi era
Husserl's career during the Nazi Germany era was full of complexity and conflict. Although he was not initially a member of the Nazi Party, he later joined the party, which affected his career both positively and negatively throughout that period. He held positions at the University of Göttingen and the University of Hamburg at different times in his career, which was a difficult time for most German academics.
A teacher who nurtures top mathematicians
Hasse was an advisor to PhD students who later made a name for themselves in their own fields, including Cahit Arf, the famous Turkish mathematician, Oswald Teichmüller, a pioneer of Teichmüller theory important for Riemann surface geometry, Paul Lorenzen, a logician and constructive mathematician, and Peter Roquette, who later continued research in number theory.
Throughout his career, Hasse was recognized as one of the most important mathematicians of the 20th century. His work permeates almost every branch of modern number theory, especially the local-global principle that made Hensel's p-adic numbers an indispensable tool in studying modern mathematics.
Helmut Hasse passed away on December 26, 1979, in Ahrensburg, near Hamburg, West Germany, at the age of 81. His academic legacy, including Hasse’s principles and the Hasse–Minkowski theorem, remains a key foundation that algebra theorists around the world use every day even now. It’s a testament to the power of connecting local analysis with universal understanding in the world of mathematics.
