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Results tagged with big-list
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user 6716
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.
19
votes
Interesting applications (in pure mathematics) of first-year calculus
An interesting application of calculus is the elementary polynomial case of Mason's ABC theorem. This yields, for instance, a completely trivial proof of the polynomial case of FLT (Fermat's Last Theo …
- 4,736
20
votes
Slick ways to make annoying verifications
Proofs exploiting universality often provide nice examples of slick ways to avoid annoying special cases. For example, the matrix identities below have trivial algebraic proofs by proceeding "generica …
53
votes
Accepted
Direct proof of irrationality?
Below is a simple direct proof that I found as a teenager:
THEOREM $\;\rm r = \sqrt{n}\;$ is integral if rational, for $\;\rm n\in\mathbb{N}$.
Proof: $\;\rm r = a/b,\;\; {\text gcd}(a,b) = 1 \implies …
- 4,736
6
votes
Good papers/books/essays about the thought process behind mathematical research
Franz Lemmermeyer's paper [1] contains a very interesting account of the truth about how Kummer was led to the invention of his ideal numbers (the popular legend is far from reality). In particular he …
2
votes
Can a mathematical definition be wrong?
This is a slightly different kind of example, namely, one where the original definition had to be revised when it later was realized that it was useless in certain contexts. Probably most readers have …
19
votes
Can a mathematical definition be wrong?
An important historical example is the difficult evolution of the correct definition of "integer" in algebraic extensions, i.e. defining algebraic integers. It was only with great difficulty that Dede …
5
votes
Are there any good nonconstructive "existential metatheorems"?
Jacobson's theorem that X^m = X rings are commutative provides an enlightening example, see my post below for further discussion and references
Abstract Thought vs Calculation
4
votes
Examples of undergraduate mathematics separation from what mathematicians should know
One might argue that (exotic) counterexamples fall into the first category. For example, when I took Munkres' course in topology it was organized around many very carefully chosen (counter)examples sh …
14
votes
Counterexamples in algebra?
Harry Hutchins "Examples of commutative rings" may be of interest.
It is based on his 1978 Chicago Ph.D. thesis under
Kaplansky, and not surprisingly it serves as a useful complement to
Kaplansky's …
7
votes
Books you would like to see translated into English
The following wonderful 54 page survey by O. Neumann on Kronecker's divisor theory could easily be turned into a book and would fill a very large gap in the English literature on such. I'm interested …
17
votes
Abstract thought vs calculation
Some of the prettiest examples of Dedekind's structuralism arise from revisiting proofs in elementary number theory from a highbrow viewpoint, e.g. by reformulating them after noticing hidden structur …
28
votes
Abstract thought vs calculation
One striking example that comes to mind is Nathan Jacobson's proof that rings satisfying the identity $X^m = X$ are commutative. This is model-theoretic and proceeds by a certain type of factorization …
29
votes
Papers that debunk common myths in the history of mathematics
One of the biggest myths in number theory is that work on Fermat's last theorem played a large role in the development of ideal theory and algebraic number theory. In fact it was much loftier goals su …