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Results tagged with big-list
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Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
32
votes
Accepted
Proofs of the Chevalley-Warning Theorem
I am working on a book-length manusript, Around the Chevalley–Warning Theorem. A complete answer to your question is estimated at about 150 pages!
In terms of what exists at the moment, here are two …
23
votes
Elementary / Interesting proofs of the Nullstellensatz
I have been thinking about the question "What is the best -- i.e., some combination of shortest, most natural, easiest -- proof of the Nullstellensatz?" recently on the eve of a commutative algebra co …
- 2,821
32
votes
Examples of theorems with proofs that have dramatically improved over time
I think that Ax's proof of the Chevalley-Warning Theorem qualifies.
The Chevalley-Warning Theorem is an affirmative solution of a conjecture made by L.E. Dickson in 1909 and taken up more seriously b …
- 4,713
85
votes
What are the most misleading alternate definitions in taught mathematics?
I increasingly abhor the introduction of the finite ring $Z_n$ not as $\mathbb{Z}/n\mathbb{Z}$ but as the set $\{0,\ldots,n-1\}$ with "clock arithmetic". (I understand that if you want to introduce m …
CommunityBot
- 1
5
votes
The best text to study both incompleteness theorems
For instance, there is a well-regarded recent book of Torkel Franzen:
Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse
A detailed and positive review was given by Panu Raatikainen in th …
- 103
9
votes
Proofs by induction
In the last year, I have twice taught a course on mathematical reasoning for future undergraduate math majors. The first time I was surprised by how conceptually difficult induction was for many of t …
CommunityBot
- 1
55
votes
Fundamental Examples
The Fermat Equation xn + yn - zn = 0.
This has truly been much more than an example in both algebra and number theory: it was one of the main motivations to develop the theory of unique factorizati …
- 65.4k
18
votes
An example of a proof that is explanatory but not beautiful? (or vice versa)
1) "There is no simple group of order $n$" (for various composite values of $n$ in the interval $[50,200] \setminus \{60,168\}$ or so). These arguments are explanatory but not beautiful. They seem v …
- 65.4k
30
votes
Consolidation: Aftermathematics of fads
Since Quinn's article is a long opinion piece which he says is 90% complete and welcomes comments, it seems entirely appropriate to contact him for clarification on this point. He would probably be h …
- 65.4k
20
votes
Awfully sophisticated proof for simple facts
I claim that the rational canonical model of the modular curve $X(1) = \operatorname{SL}_2(\mathbb{Z}) \backslash \overline{\mathcal{H}}$ is isomorphic over $\mathbb{Q}$ to the projective line $\mathb …
- 65.4k
7
votes
Collecting proofs that finite multiplicative subgroups of fields are cyclic
I actually think it will not be so easy to say when two proofs of this result will be "distinctly different": rather I expect most or all will have common features, including using at least a little b …
CommunityBot
- 1
20
votes
Interesting Calculus Questions/Exercises
I have little personal experience with it, but some colleagues and friends hold the following text in high regard:
Robert M. Young, Excursions in calculus.
An interplay of the continuous and the …
- 65.4k
2
votes
Applications of periodic continued fractions
There is a pleasant connection between (among?) Chebyshev polynomials, the Pell equation and continued fractions, the latter two being understood to take place in real quadratic function fields rather …
- 65.4k
41
votes
How to present mathematics to non-mathematicians?
For some reason, many mathematicians have trouble with the idea that when some layman asks them about their work, the appropriate response is not to try to figure out how to describe the latest theore …
- 65.4k
52
votes
Most memorable titles
Finding composite order ordinary elliptic curves using the Cocks-Pinch method, by D. Boneh, K. Rubin and A. Silverberg. (To appear in the Journal of Number Theory.)
- 65.4k