mathematics

Philipp Furtwängler

Philipp Furtwängler 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 Philipp Furtwängler rather than just read about it. In short: 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 zerlegbaren ganzzahligen ternären kubischen Formen), under Felix Klein.

Key takeaways

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

Reference excerpt

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 zerlegbaren ganzzahligen ternären kubischen Formen), under Felix Klein. Most of his academic life, from 1912 to 1938, was spent at the University of Vienna, where he taught for example Kurt Gödel, who later said that Furtwängler's lectures on number theory were the best mathematical lectures that he ever heard; Gödel had originally intended to become a physicist but turned to mathematics partly as a result of Furtwängler's lectures. From 1916, Furtwängler became increasingly paralysed and, without notes, lectured from a wheelchair while his assistant wrote equations on the blackboard. Some of Furtwängler's doctoral students were Wolfgang Gröbner, Nikolaus Hofreiter, Henry Mann, Otto Schreier, and Olga Taussky-Todd. Through these and others, he has over 3000 academic descendants. He is now best known for his contribution to the principal ideal theorem in the form of his Beweis des Hauptidealsatzes für Klassenkörper algebraischer Zahlkörper (1929). Philipp Furtwängler was a grandson of the organ builder Philipp Furtwängler (1800-1867) and a second cousin of the conductor Wilhelm Furtwängler.

Selected publications with Helmut Hasse and W. Jehne: Allgemeine Theorie der algebraischen Zahlen. Vol. 8. Teubner, 1953.

See also Eisenstein reciprocity Hilbert class field Keller's conjecture Kummer–Vandiver conjecture Principalization (algebra)

References

Sources "Philipp Furtwängler". In: Österreichisches Biographisches Lexikon 1815–1950 (ÖBL). Vol. 1, Austrian Academy of Sciences, Vienna 1957, p. 383. Nikolaus Hofreiter (1961). "Furtwängler, Friedrich Pius Philipp". Neue Deutsche Biographie (in German). Vol. 5. Berlin: Duncker & Humblot. pp. 740–740.

External links Literature by and about Philipp Furtwängler in the German National Library catalogue http://bibliothek.bbaw.de/kataloge/literaturnachweise/furtwaen/literatur.pdf Archived 2016-05-28 at the Wayback Machine (PDF-Datei; 35 kB) Friederich Pius Philipp Furtwängler at the MacTutor History of Mathematics archive

Illustrations

Worked examples

Example 1 — a first encounter with Philipp Furtwängler

Start with the simplest possible case. Write down what Philipp Furtwängler 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 Philipp Furtwängler 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 Philipp Furtwängler 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 Philipp Furtwängler

In research: Philipp Furtwängler 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 Philipp Furtwängler 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: Philipp Furtwängler is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1869 births, 1940 deaths, 19th-century German mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Philipp Furtwängler 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 Philipp Furtwängler in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Philipp Furtwängler 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.
Explain Philipp Furtwängler out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Philipp Furtwängler in simple terms?

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 zerlegbaren ganzzahligen ternären kubischen Formen), under Fel…

Why does Philipp Furtwängler 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 Philipp Furtwängler?

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 Philipp Furtwängler.