# Mathematicians in History: Hans Julius Zassenhaus

**Aug. 26, 2009**

Hans Zassenhaus was a great mathematician of the Twentieth Century. The purpose of this article is to introduce you to his life and work. Zassenhaus was born May 28, 1912 in Koblenz-Moselweiss, Germany, a small town near the confluence of the rivers Mosel and Rhein. This was 40 years before and some 60 miles north of where yours truly drew his first breath. His family soon moved to the big port city of Hamburg. In 1930 he entered the University of Hamburg to study mathematics and physics with an emphasis on the physics of the atom. As easily happens to a young man in college, he soon fell in with the wrong (or right?!) crowd, which in this case consisted of the world famous algebraists Emil Artin and Erich Hecke. The great Artin became Zassenhaus' major professor and supervised his dissertation research on sharply 3-transitive permutation groups, which are now called Zassenhaus groups. Hans Zassenhaus received his doctorate in 1934, at the tender age of 22. Let me hasten to say that in no way, shape or form am I comparing myself to Zassenhaus. I just find it mildly amusing that I entered the University of Kaiserslautern in 1970 and got my doctorate in 1974, again a perfect 40 year time shift.

Zassenhaus soon became Artin's assistant professor and kept that position even after Artin was fired by the Nazis in 1937 because he had a Jewish wife. Artin went to Indiana and Zassenhaus stayed behind in Hamburg. In 1940 the Nazi government insisted that all university faculty be members of the Nazi party. Zassenhaus did not succumb to the Zeitgeist. He quit his position and joined the German navy, where he worked as a weather forecaster until the end of World War II. After the war, he could have become a professor at the prestigious university at Bonn, but he declined, insisting that the position be filled by somebody whose job had been lost under Nazi rule. Zassenhaus left Europe in 1949 (as I did 35 years later) and went to Montreal, Canada. He moved to Notre Dame in 1959 and eventually to Ohio State University in 1963, where he stayed until his retirement in 1981. He died in Columbus, Ohio on November 21, 1991. His list of publications has 188 items, and he had about 40 students.

Zassenhaus' work had a huge impact in many areas of mathematics. He made lasting contributions to group theory, crystallography, nearfields and their applications to geometry, the theory of orders, representation theory, theoretical and computational(!) algebraic number theory, the geometry of numbers, (modular) Lie algebras and their application in physics, didactics and history of mathematics. At the time when the Bourbaki school ruled and everything in mathematics became abstract and axiomatized, Zassenhaus developed computer algorithms to compute the invariants of algebraic number fields, like their Galois groups, etc. As early as 1959 he pioneered the use of computers to run experiments in algebra, and he developed algorithms to factor polynomials and much more. At a time when research was valued much more than teaching, Zassenhaus was a passionate teacher, who liked to teach mathematics from a historical perspective. He even wrote articles on the teaching of High School algebra, bucking the Zeitgeist once more. He was a perfect model of Baylor University's paradigm of a scholar/teacher! In 1963, A. L. S. Corner († 2006) published his celebrated result that each ring R whose additive group (R,+) is torsion-free and reduced of finite rank n can be realized as the endomorphism ring of some abelian group A of rank 2n. In a little noticed paper in 1967, Zassenhaus proved, in a constructive manner, that if (R,+) is free of finite rank n, then A can be constructed to have rank n only. Actually, A is sandwiched between (R,+) and its divisible hull. Working with my student, J. Buckner, such groups A popped up again all of a sudden as localizations of the ring of integers in the quasi-category of abelian groups, whatever that may be. During a recent sabbatical, I worked once again with Dr. Gobel and we generalized Zassenhaus' result to certain rings of infinite rank. This paper has appeared in Fundamenta Mathematicae.

At times there has been debate about the role of experiment in mathematics and the importance of proof. Zassenhaus had this to say: "Indem wir den Mut zum Experimentieren beweisen, werden uns Experimente den Mut zum Beweisen geben." Translation: By proving our courage to experiment, the experiments will give us the courage to prove.

References:

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zassenhaus.html

http://www.math.ohio-state.edu/node/19548/print

In the paper with Gobel mentioned above, we called rings that satisfy the conclusion of Zassenhaus' result "Zassenhaus rings." We hope that this will stick.

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