mathematics

Otto Schilling

Otto Schilling is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Otto Schilling rather than just read about it. In short: Otto Franz Georg Schilling (3 November 1911 – 20 June 1973) was a German-American mathematician known as one of the leading algebraists of his time. He was born in Apolda and studied in the 1930s at the Universität Jena and the Universität Göttingen under Emmy Noether.

Key takeaways

Otto Schilling belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Otto Schilling to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Otto Schilling from memory before moving on to harder problems.

Reference excerpt

Otto Franz Georg Schilling (3 November 1911 – 20 June 1973) was a German-American mathematician known as one of the leading algebraists of his time. He was born in Apolda and studied in the 1930s at the Universität Jena and the Universität Göttingen under Emmy Noether. After Noether was forced to leave Germany by the Nazis, he found a new advisor in Helmut Hasse, and obtained his Ph.D. from Marburg University in 1934 on the thesis Über gewisse Beziehungen zwischen der Arithmetik hyperkomplexer Zahlsysteme und algebraischer Zahlkörper. He then was post doc at Trinity College, Cambridge before moving to Institute for Advanced Study 1935–37 and the Johns Hopkins University 1937–39. He became an instructor with the University of Chicago in 1939, promoted to assistant professor 1943, associate 1945 and full professor in 1958. In 1961 he moved to Purdue University. He died in Highland Park, Illinois. His students were, among others, the game theorist Anatol Rapoport and the mathematician Harley Flanders.

Articles Schilling, Otto F. G. (1937). "Arithmetic in a Special Class of Algebras". The Annals of Mathematics. 38 (1): 116–119. doi:10.2307/1968513. ISSN 0003-486X. JSTOR 1968513. Schilling, Otto F. G. (1937). "Class Fields of Infinite Degree Over p-Adic Number Fields". The Annals of Mathematics. 38 (2): 469–476. doi:10.2307/1968563. ISSN 0003-486X. JSTOR 1968563. Schilling, O. E. G. (1937). "Arithmetic in Fields of Formal Power Series in Several Variables". The Annals of Mathematics. 38 (3): 551–576. doi:10.2307/1968600. ISSN 0003-486X. JSTOR 1968600. (typo in Schilling's name) Schilling, O. F. G. (1938). "The Structure of Local Class Field Theory". American Journal of Mathematics. 60 (1): 75–100. doi:10.2307/2371544. ISSN 0002-9327. JSTOR 2371544. Schilling, O. F. G. (1938). "A Generalization of Local Class Field Theory". American Journal of Mathematics. 60 (3): 667–704. doi:10.2307/2371605. ISSN 0002-9327. JSTOR 2371605. Schilling, O. F. G. (1939). "Units in p-Adic Algebras". American Journal of Mathematics. 61 (4): 883–896. doi:10.2307/2371632. ISSN 0002-9327. JSTOR 2371632. with Saunders Mac Lane: MacLane, Saunders; Schilling, O. F. G. (1939). "Zero-Dimensional Branches of Rank One on Algebraic Varieties". The Annals of Mathematics. 40 (3): 507–520. doi:10.2307/1968935. ISSN 0003-486X. JSTOR 1968935. with Saunders Mac Lane: Lane, Saunders Mac; Schilling, O. F. G. (1939). "Infinite Number Fields with Noether Ideal Theories". American Journal of Mathematics. 61 (3): 771–782. doi:10.2307/2371335. ISSN 0002-9327. JSTOR 2371335. with Saunders Mac Lane: MacLane, S.; Schilling, O. F. G. (1940). "Normal Algebraic Number Fields". Proceedings of the National Academy of Sciences. 26 (2): 122–126. doi:10.1073/pnas.26.2.122. ISSN 0027-8424. PMC 1078017. PMID 16588322. Schilling, O. F. G. (1940). "Regular normal extensions over complete fields". Trans. Amer. Math. Soc. 47 (3): 440–454. doi:10.1090/s0002-9947-1940-0001970-2. MR 0001970. Schilling, O. F. G. (1940). "Remarks on a Special Class of Algebras". American Journal of Mathematics. 62 (1/4): 346–352. doi:10.2307/2371458. ISSN 0002-9327. JSTOR 2371458. with Saunders Mac Lane: MacLane, Saunders; Schilling, O. F. G. (1942). "A formula for the direct products of crossed product algebras". Bull. Amer. Math. Soc. 48 (2): 108–114. doi:10.1090/s0002-9904-1942-07613-0. MR 0006152. with Irving Kaplansky: Kaplansky, Irving; Schilling, O. F. G. (1942). "Some remarks on relatively complete fields". Bulletin of the American Mathematical Society. 48 (10): 744–748. doi:10.1090/S0002-9904-1942-07772-X. ISSN 0002-9904. Schilling, O. F. G. (1943). "Normal Extensions of Relatively Complete Fields". American Journal of Mathematics. 65 (2): 309–334. doi:10.2307/2371818. ISSN 0002-9327. JSTOR 2371818. Schilling, O. F. G. (1945). "On a special class of abelian functions". Bull. Amer. Math. Soc. 51 (2): 133–136. doi:10.1090/s0002-9904-1945-08293-7. MR 0011291. Schilling, O. F. G. (1945). "Noncommutative valuations". Bull. Amer. Math. Soc. 51 (4): 297–304. doi:10.1090/s0002-9904-1945-08339-6. MR 0011684. Schilling, O. F. G. (1946). "Ideal theory on open Riemann surfaces". Bull. Amer. Math. Soc. 52 (11, Part 1): 945–963. doi:10.1090/s0002-9904-1946-08669-3. MR 0019733. "Necessary conditions for local class field theory" (PDF). Mathematical Journal of Okayama University. 3 (1): 5–10. 1953. ISSN 0030-1566. f. g. Schilling, O. (1961). "On local class field theory". Journal of the Mathematical Society of Japan. 13 (3): 234–245. doi:10.2969/jmsj/01330234. ISSN 0025-5645.

Books Schilling, Otto (1950). Theory of Valuations. American Mathematical Society. ISBN 0821815040. {{cite book}}: ISBN / Date incompatibility (help) Schilling, Otto; Piper, William Stephen (1975). Basic abstract algebra. Allyn & Bacon. ISBN 0205042732; (394 pages){{cite book}}: CS1 maint: postscript (link)

References

Worked examples

Example 1 — a first encounter with Otto Schilling

Start with the simplest possible case. Write down what Otto Schilling claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Otto Schilling before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Otto Schilling ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Otto Schilling

In research: Otto Schilling appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Otto Schilling in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Otto Schilling is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1911 births, 1973 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Otto Schilling outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Otto Schilling in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Otto Schilling means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
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Frequently asked questions

What is Otto Schilling in simple terms?

Why does Otto Schilling matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Otto Schilling?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Otto Schilling.