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Arnold Scholz: Between Mathematics and Politics

This article presents an English translation and modified introduction to the 2016 correspondence between Hasse, Scholz, and Taussky, offering insights into the life and work of the number theorist Arnold Scholz.

Original authors: Franz Lemmermeyer

Published 2026-02-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Franz Lemmermeyer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Story of a Mathematical Genius in a Dark Time

Imagine a brilliant architect named Arnold Scholz. He lived in Germany in the early 20th century. His job wasn't to build houses, but to build mathematical worlds. Specifically, he was an expert in "Number Theory," which is like the study of the DNA of numbers.

This paper tells the story of two things happening at the same time:

The Math: How Scholz built incredible, complex structures that other mathematicians didn't fully understand yet.
The Politics: How the rise of the Nazi party in Germany destroyed his career, isolated him from his friends, and eventually led to his tragic death.

Part 1: The Mathematical Architect (The "Goethe" vs. "Landau" Problem)

Scholz was a genius, but he had a unique way of thinking. The paper compares him to two famous German figures: Goethe (a poet who wrote in flowing, emotional language) and Landau (a mathematician who wrote in strict, rigid, logical steps).

Scholz was "Too much Goethe, too little Landau."
- The Analogy: Imagine Scholz trying to explain how to build a castle. Instead of giving you a step-by-step blueprint (1. lay bricks, 2. add mortar), he would say, "Just look at the castle in your mind's eye; you can see the colors and the shapes."
- The Result: He could "see" the answers instantly. But when he wrote them down, they were confusing. He didn't break his ideas into small, clear rules (lemmas). He hid the rules inside the story.
- The Consequence: For decades, other mathematicians read his papers and thought, "This is brilliant, but I can't follow it." They would rediscover his ideas years later, not realizing Scholz had already solved them. It's like finding a treasure map written in a secret code that no one could crack until much later.

His Big Mathematical Ideas:

The "Knot" Theory: Scholz discovered that numbers have "knots." Imagine a rope connecting different mathematical worlds. Sometimes, the rope is tight (a knot exists), and sometimes it's loose. He figured out how to tie and untie these knots to build new number systems.
The "Wreath Product": This is a fancy way of stacking groups of numbers on top of each other. Scholz realized you could build complex towers of numbers by stacking simple blocks, provided you followed a specific pattern.
The "Reflection Theorem": He found a mirror. If you look at a number field in a mirror (a specific mathematical transformation), the number of "holes" (class numbers) in the reflection is almost the same as in the original. This was a huge discovery that helped solve puzzles about prime numbers.

Part 2: The Political Storm (The "Bad Neighbor" Problem)

While Scholz was building his mathematical towers, a storm was brewing outside his window.

The "Bad Neighbors" (The Nazis): In 1933, the Nazi party took over Germany. They had a strict rule: "No Jews allowed."
The "Jewish" Label: Scholz wasn't Jewish, but he was friends with Jewish mathematicians (like Emmy Noether and Issai Schur). He also refused to join the Nazi student groups or say the "Heil Hitler" salute.
The "Snitch" (Doetsch): Scholz had a boss named Gustav Doetsch. Doetsch was a "useful idiot" for the regime. He wanted to get ahead, so he turned on Scholz.
- The Analogy: Imagine a teacher who loves math but hates the new principal. The teacher (Scholz) just wants to teach. But the principal's favorite assistant (Doetsch) tells the principal, "That teacher is dangerous! He hangs out with the wrong crowd and won't salute the flag."
- The Result: Doetsch wrote nasty letters calling Scholz a "whiner" and a "communist sympathizer." He blocked Scholz from getting jobs, even though Scholz was one of the best mathematicians in the country.

The Tragic Journey:

Freiburg: Scholz was forced out of his job in Freiburg because his boss (Loewy) was Jewish and fired. Scholz was the assistant, so he lost his job too.
Kiel: He moved to Kiel, but the Nazis there were still watching him. He was promised a job, but it was delayed for months. He was starving for work and money.
Jena: He finally got a job in Jena, but the political pressure followed him.
The War: When World War II started, Scholz was drafted. He wasn't a soldier; he was a radio operator and later a teacher at a naval academy. He was exhausted, depressed, and lonely.

Part 3: The Tragic End

Scholz died in 1942 at the age of 37.

The Cause: The paper says he died of "galloping diabetes." This is an old term for a disease that attacks you very fast.
The Metaphor: Think of Scholz as a brilliant lighthouse keeper. He was trying to keep the light of mathematics shining during a time of total darkness (war and fascism). But the storm was too strong. The political winds blew out his light, and he died young, broken by the stress and the loss of his friends.

The Aftermath:

Lost Treasure: When Scholz died, his mother tried to send his notes and papers to relatives in Poland. But during the war, the Germans were expelled from Poland, and all of Scholz's life work was lost. It's like a library burning down, taking thousands of books with it.
Rediscovery: Decades later, mathematicians like Olga Taussky (his friend) and Wolfram Jehne started reading his old, messy notes again. They realized, "Wait! He figured this out 50 years ago!" They finally translated his "Goethe-style" poetry into modern "Landau-style" math.

Summary: The Lesson

The paper is a tribute to a man who was too smart for his own good in a stupid world.

Mathematically: He was a visionary who saw patterns others missed, but he couldn't explain them clearly enough to be understood in his time.
Politically: He was a victim of hatred. He refused to compromise his integrity or his friendships, and the system crushed him.

The author, Franz Lemmermeyer, is essentially saying: "We owe it to Arnold Scholz to finally translate his genius into a language we can all understand, so we don't lose his brilliant ideas to history."

Based on the provided text, here is a detailed technical summary of Franz Lemmermeyer's article, "Arnold Scholz: Between Mathematics and Politics."

1. Problem and Context

The paper addresses two intertwined problems:

Mathematical: To reconstruct and clarify the number-theoretic work of Arnold Scholz (1904–1942), whose results were often difficult to read due to his unique, non-formal style and the lack of modern algebraic language (specifically cohomology theory) during his lifetime. The author aims to "translate" Scholz's insights into modern terminology to highlight his priority in discovering concepts later rediscovered by others.
Historical/Biographical: To document the career trajectory of Scholz under the National Socialist regime in Germany, detailing how political persecution, the dismissal of Jewish colleagues (like Issai Schur and Alfred Loewy), and the rise of Nazi ideologues in academia (like Gustav Doetsch and Wilhelm Süss) obstructed his career and ultimately led to his premature death.

2. Methodology

Lemmermeyer employs a historical-biographical and mathematical reconstruction approach:

Chronological Narrative: The paper follows Scholz's life from 1927 to 1942, interspersing biographical events with his mathematical output.
Technical Translation: The author re-expresses Scholz's original proofs and definitions using modern algebraic number theory, specifically:
- Cohomology: Translating Scholz's "knots" (Knoten) into the language of Galois cohomology ( $H^2$ ) and Schur multipliers.
- Group Theory: Clarifying Scholz's concepts of "disposition groups" (wreath products) and "branch groups" (maximal $p$ -groups).
- Class Field Theory: Re-interpreting Scholz's work on norm residues, capitulation, and class field towers.
Archival Analysis: The author utilizes correspondence between Scholz, Helmut Hasse, and Olga Taussky, as well as Nazi-era administrative files (e.g., Doetsch's denunciations), to reconstruct the political pressures Scholz faced.

3. Key Contributions and Mathematical Results

A. Inverse Galois Problem and Group Construction

Disposition Groups: Scholz identified that constructing extensions with solvable Galois groups could be reduced to constructing extensions with specific "disposition groups" (modern term: wreath products $C_m \wr N$ ) and branch groups.
Ramification Control: He developed methods to construct extensions with prescribed Galois groups where ramification is controlled (often occurring at exactly one prime ideal per step).
Schur Multipliers: Scholz implicitly utilized the Schur multiplier (though not named as such in his early work) to solve embedding problems. He showed that the embeddability of certain fields into non-abelian extensions is linked to the non-triviality of the Schur multiplier of the Galois group.

B. Class Field Towers and Rank

Infinite Towers: In 1929 ([S5]), Scholz proved that cyclic number fields of prime degree $\ell$ possess $\ell$ -class field towers of arbitrarily great length, provided the rank of the $\ell$ -class group is sufficiently large. This was a precursor to the definitive Golod-Shafarevich theorem (1964) proving the existence of infinite class field towers.
Capitulation: Working with Olga Taussky, he investigated the capitulation of ideal classes in unramified extensions, linking the kernel of the capitulation map to cohomological groups ( $H^1$ ).

C. The Theory of Knots (Number Knots)

Definition: Scholz introduced the concept of "knots" to describe the discrepancy between global norms and local norm residues.
- Number Knot ( $\nu$ ): $N/NL/K L^\times$
- Unit Knot ( $\omega$ ): $E_K \cap N / N E_L$
- Ideal Knot ( $\delta$ ): Related to principal ideals.
Significance: He established exact sequences relating these knots (e.g., $1 \to \omega \to \nu \to \delta \to 1$ ). Lemmermeyer notes that these correspond to the Schur multiplier $M(G) \cong H^2(G, \mathbb{Q}/\mathbb{Z})$ and the Tate-Shafarevich group of the norm torus. This work was largely ignored until Wolfram Jehne modernized it in the 1970s.

D. Reciprocity Laws and Units

Scholz Reciprocity Law: He generalized Schönumann's law regarding the quartic residue symbol of fundamental units. Specifically, for primes $p, q \equiv 1 \pmod 4$ , he established a reciprocity relation for the fundamental units $\varepsilon_p$ and $\varepsilon_q$ .
Reflection Theorem: He proved a theorem relating the 3-ranks of the class groups of quadratic fields $K = \mathbb{Q}(\sqrt{m})$ and $K' = \mathbb{Q}(\sqrt{-3m})$ , showing their ranks differ by at most 1. This was later generalized by Leopoldt.

E. Minimal Discriminants

Scholz investigated the minimal discriminants of number fields of degree $n$ . He conjectured that the root discriminant $rd(K) = |D_K|^{1/n}$ could be made arbitrarily small relative to $n$ (i.e., $\liminf rd(K)/n = 0$ ) by constructing specific towers of fields, a problem that remains open.

4. Results and Historical Findings

Priority of Discovery: The paper demonstrates that Scholz discovered many results decades before they were "rediscovered" by others (e.g., Shafarevich on solvable groups, Jehne on knots, Lehmer on reciprocity laws).
Political Persecution:
- Scholz was a "non-Aryan" sympathizer (associated with Jewish colleagues like Schur, Noether, and Taussky) and refused to align with Nazi ideology.
- He was dismissed from his assistant position in Freiburg in 1934 by the Nazi-aligned Doetsch, who labeled him a "grumbler" and "unreliable."
- Despite offers from Hasse and Schmidt, he struggled to find a permanent position in Germany due to political vetting. He eventually moved to Kiel and later Jena but remained precarious.
Death: Scholz died in 1942 at age 37 from "galloping diabetes" (aggressive diabetes) while serving as a civilian teacher at a naval school in Flensburg. His estate and manuscripts were lost during the expulsion of Germans from Poland in 1945.

5. Significance

Mathematical Legacy: The paper argues that Scholz was a visionary who anticipated the language of cohomology and group cohomology in number theory. His "knots" are now recognized as fundamental in the study of embedding problems and the arithmetic of tori.
Historical Correction: It corrects the historical record by highlighting the loss of a brilliant mathematician due to the Nazi regime. It illustrates how the "Jewish influence" purge in German mathematics destroyed not only Jewish scholars but also those who worked with them or opposed the regime.
Pedagogical Value: By translating Scholz's "spatial image" thinking into formal lemmas and propositions, Lemmermeyer makes Scholz's profound but obscure insights accessible to modern researchers, bridging the gap between early 20th-century intuition and modern algebraic rigor.

In summary, the paper serves as both a mathematical rehabilitation of Arnold Scholz's work, placing him as a central figure in the development of class field theory and inverse Galois theory, and a historical testament to the destruction of German mathematics under National Socialism.