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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.
260
votes
Accepted
Consequences of the Riemann hypothesis
I gave a talk on this topic a few months ago, so I assembled a list then which could be appreciated by a general mathematical audience. I'll reproduce it here. (Edit: I have added a few more examples …
- 50.6k
48
votes
Collecting proofs that finite multiplicative subgroups of fields are cyclic
I once collected six [edit: now seven [edit: now eight [edit:now nine [edit: now ten]]]] proofs of this theorem, for the field $\mathbf Z/(p)$, and they can be found here. While $\mathbf Z/(p)$ is not …
- 50.6k
35
votes
Applications of the Chinese remainder theorem
Here are some applications I don't see listed among the other answers.
Everyone knows $5^2$ ends in 5 and $6^2$ ends in 6. Your task: find multi-digit numbers whose squares end in themselves (e.g., $ …
- 8,842
53
votes
Do you read the masters?
In algebraic number theory, the existence of a Frobenius element
at any prime $p$ in a Galois extension $K/{\mathbf Q}$ is crucial. That is, for any
prime ideal $\mathfrak p$ lying over $p$ in $K$ th …
- 4,713
9
votes
Differences between $p$-groups and $q$-groups
If it exists, what is a value $k$ for which there are different numbers of (isomorphism classes of) groups of order $p^k$ and $q^k$?
You already observed that $k$ can't be $1$, $2$, or $3$. So such …
- 50.6k
27
votes
Swimming against the tide in the past century: remarkable achievements that arose in contras...
In the first decades of the 20th century, $p$-adic analysis (or valuation theory more generally) was regarded by many as rather exotic. After Hensel's work there was a steady development by Strassmann …
- 12.8k
23
votes
Modern results that are widely known, yet which at the time were ignored, not accepted or cr...
Does acceptance of conjectures before they became theorems count?
Example 1. The Artin reciprocity law. When Artin went around to other people describing what he was trying to show, nobody else belie …
- 50.6k
6
votes
What well known results with countability assumptions can be naturally extended to uncountab...
Here are some examples from algebra where finiteness assumptions can be removed. In the first two, the statement of the more general result is unchanged, but the third result has to be expressed in a …
- 50.6k
24
votes
How would you have answered Richard Feynman's challenge?
Here are two questions, and both are about math that was known long before Feynman passed away.
Explain to him what unique factorization into irreducibles means (including the ambiguity from multipli …
- 50.6k
21
votes
Widely accepted mathematical results that were later shown to be wrong?
Any rational function field over a finite field has genus $0$ and class number $1$, where the class number of a function field over a finite field is the number of degree-zero elements of the divisor …
- 4,713
227
votes
Widely accepted mathematical results that were later shown to be wrong?
Mathematicians used to hold plenty of false, but intuitively reasonable, ideas in analysis that were backed up with proofs of one kind or another (understood in the context of those times). Coming to …
- 50.6k
15
votes
Ways to prove the fundamental theorem of algebra
Here is a translation into English of a second "real" proof from the journal Ilya mentioned in his answer. This proof is due to Petya Pushkar', in the 1997 paper titled О некоторых топологических док …
- 35.5k
28
votes
Examples of improved notation that impacted research?
There is a notation that had an immediate and profound impact on research in algebraic topology, later algebraic geometry, and was eventually adopted by all areas of mathematics: the introduction of a …
- 50.6k
16
votes
Noteworthy, but not so famous conjectures resolved recent years
In number theory, the Sato-Tate conjecture about elliptic curves over $
\mathbf Q$ was a problem from the 1960s and Serre's conjecture on modularity of odd 2-dimensional Galois representation was a co …
- 50.6k
11
votes
Shortest/Most elegant proof for $L(1,\chi)\neq 0$
There are proofs that treat the cases of real and non-real $\chi$ on an equal footing. One proof is in Serre's Course in Arithmetic, which the answers by Pete and David are basically about. That met …
- 8,842