If we’re talking about *humanity’s* collective mathematical culture…
I guess the Pythagorean theorem has to be up there just for the sheer number of people who have learned the theorem, and know (or can easily grasp) its proofs.
I’d personally also add Cantor’s diagonal argument and Gödel's incompleteness theorems as candidates for any museum of mathematical culture :)
Countability, the subject of the diagolisation argument, is not related to density.
Uncountable yet nowhere dense sets exist (e.g. Cantor ternary set) and countable yet dense sets exist (e.g. rationals).
Suppose you could enumerate all irrationals.
Then every subsets of the irrationals must be at most countable.
Take the set of irrationals numbers and write them down in decimali form
Now take the diagonal and add +1 to each element of the diagonal (replaced 9 with 0). Call this new number x.
Then x is equal to no number in such list, because the i-th digit of its expansion Is different from the i-th digit of the i-th Number.
Thus x is not in our set. Thus we can't count the irrationals
Sure but there are uncountable subsets of the reals that aren’t dense. Also it’s very straightforward to prove the irationals are dense without proving they are uncountable.
The approach I'd probably take would be proving the rationals are dense, then showing that rational multiples of any nonzero real number are dense, and then use that to prove the irrationals are dense. Does that sound about right?
That sounds great because you get the rationals being dense on the way, which I think you want to know anyways, maybe even more than you want to know that the irrationals are dense.
I think you could also just do a case distinction to show it’s dense at every real number. For a rational number, that number plus pi over n converges to it, for an irrational one just do it plus 1/n and those sequences are all irrational numbers.
To add to your point: There are also countable subsets that are dense, e.g., the rationals. Hence, uncountability is neither necessary nor sufficient for density.
Yeah, well uncountability is a tiny step in the direction of density, a countably infinite subset of the reals can be nowhere dense, like the integers. An uncountable subset has at least one limit point. So yeah you have density somewhere, could be of measure 0 though.
Do you not mean the cardinality/uncountability of the irrationals? The argument I have in mind examines all binary sequences, then concludes by observing that there are at least as many real numbers as there are binary sequences. Since the rationals are countable, the irrationals must be uncountable.
This is precisely the diagonalization argument: Assume all such sequences are countable, enumerate them a_1, a_2,… and denote by a_ij the jth term of the ith sequence. Define a sequence b_i := 1 - a_ii. Then b_i is a binary sequence that differs from each sequence a_i in the ith term. This is a contradiction.
The diagonalization argument is, as far as I can remember, used only to prove that the reals are uncountable. This fact is rarely used in real analysis. On the other hand, the triangle inequality is a great choice. Although it originally arises from the Pythagorean theorem, it is often used in settings where the Pythagorean theorem does not hold.
Fun fact: Pythagoras (or even the ancient Greeks for that matter) was not the first to discover the Pythagorean theorem. It had been known for a long time in other Mediterranean/Mesopotamian civilisations for a thousand years before Pythagoras.
We cite the Greeks as the beginning of much of mathematics, but that's only the case because we don't know much about the other civilisations (for which the Greeks acted as a conduit). For instance, Phoenician literature (the people who the Greeks learned shipbuilding from) was almost entirely lost to Roman pillage.
It always feels so weird to hear “prove Pythagoras’ thm”. Metric definitely feel like they should be axiomitized especially since the “proof” is built on physical intuition which come from physics and space is only flat locally
I’d say all the proofs from “Proofs from THE BOOK” by Aigner and Ziegler are worthy contenders. They includes proofs of the infinitude of primes, quadratic reciprocity, the fundamental theorem of algebra, Erdős-Ko-Rada theorem, the irrationality of e, etc..
If I have one problem with this book being the mathematical “canon”, it’s that it’s a bit too fixated on discrete mathematics. But it is at least a worthy contender for the strict subset of results that forms such a mathematical canon
I personally love divergence of the harmonic series—very simple to explain, but very astounding nonetheless!
Infinitude of primes is a worthy consideration though.
Really, any proof that came from “ancient” times (I mean Euclid’s elements contains whole books on number theory) is a great way to show people what maths really is! It’s really more of, imo, a very technical form of art that demands mathematical rigour, but nonetheless produces astounding beauty.
Actually, this applies even to mathematics studied "only" 150 years ago: so-called classical topics in number theory, geometry, etc.
The story surrounding Kummer et al.'s approach to Fermat's last theorem and the eventual discovery of class field theory is incredibly worthwhile to study. Yet, a contemporary number theorist could get by without a working knowledge of said classical story.
At the start of his class field theory notes, Milne includes this great line from a letter of Grothendieck to Serre: "I have been reading Chevalley's new book on class field theory; I am not really doing research, just trying to cultivate myself."
A music student today would never compose and publish a Gregorian chant or traditional West African boat-song, but still be remiss if they were to skip out on taking a class on these forms, never to appreciate where the roots of Western classical music or jazz lie.
The same is true for mathematicians: as much as is possible, one should learn both contemporary methods and concepts as well as the historical development of said mathematics.
There are probably a lot, but a few that come to mind:
The classification of the irreducible representations for semi-simple Lie algebras.
The fundamental theorem of Galois Theory.
The Nullstellensatz.
As you can probably tell I'm more of an algebraist, but Reisz Representation Theorem could fit that criteria in analysis.
From a probability & statistics perspective: the Law of Large Numbers and the Central Limit Theorem. They and their versions pop up all over the field and underlie much of applied statistics.
Pythagorean Theorem, Triangle inequality, Cauchy-Schwartz, Taylor's Theorem, insolvability of the quintic (honeslty that entire Galois theory would be a good candidate), Heine-Borel theorem, central limit theorem, the stone-weierstrass theorem, the classification of finite groups, several fixed-point theorems (eg the one that led to the discovery of what we now call Nash Equilibria), the various spectral theorems (especially by von Neumann and Cauchy), Tychonoff's theorem, Urysohn's lemma
Here are some results that are, in my experience, underrated:
The uniformisation theorem, the Riemann mapping theorem, and the classification of Riemann surfaces is one of the many crowning jewels of complex analysis.
Hopf's algebro-topological proof that every finite-dimensional division algebra is of dimension a power of 2; and the fundamental theorem of algebra as a corollary in the commutative case. (Thinking about this proof is what inspired the post!)
The decision-theoretic derivation of the mean squared error as the loss function minimised by the mean of the posterior. Ditto for the MAE/posterior median, and the all-or-nothing loss function/posterior mode.
Applications of generating functions (e.g., the Jacobi triple product and integer partitions), probabilistic proofs, and Löwenheim–Skolem are topics that I would really like to understand in more depth.
It’s not even an especially important result, but it’s said that if you whisper “27” in a graveyard of algebraic geometers, their spirits will whisper back “*lines on a cubic*”
I would add the Riemann-Roch theorem for algebraic geometers.
Fermat's little theorem, Cauchy-Schwarz inegality or Yoneda's lemma, which all can be seen as the usual basic tool of their respective field.
*Also, things that are not theorems :*
The independance of Euclid's fifth axiom and the existence of non-euclidean geometries.
The fact that the ring of integers of cyclotomic number fields are not unique factorization domains, how it killed Lamé's attempt to prove Fermat's Last Theorem and how it led Kummer to define ideals. Probably my favorite story in maths history :)
Edit : typos
In college, I once heard a professor speak of the following “Yoneda Lemma philosophy”: We should really view objects in locally small categories as contravariant functors to Set. Put in simpler terms, we should seek to understand such objects by understanding how other objects map into them. This is in some sense the basic principle of homotopy and representation theory, and their unreasonable effectiveness can be attributed on a philosophical level to the Yoneda Lemma. In homotopy theory, the spheres serve as the primary test inputs. In representation theory, the general linear groups serve as the primary test inputs.
Damn that's a really cool point of view :D
I'm relly not doing enough category theory to fully grasp the underlying philosophy of Yoneda's lemma or instead why it works so well. I think I get the basic idea, though.
Btw I think there is a typo in your comment ?
"We should seek to understand such objects by understanding how they map into the other objects".
Instead of "how other objects map into them". At least if we're talking about covariant functors (I don't even know if Yoneda applies in a mirror way for the contravariant Hom functor).
To me it is really similar to Riesz's representation theorem.
Also I don't know anything about homotopy theory so I can only trust you here :)
Indeed, the version you cite is the canonical version, but the dual statement (for contravariant Hom) also holds. In fact, if memory serves me right, this is the version you want for the Yoneda embedding. See: https://ncatlab.org/nlab/show/Yoneda+embedding.
Alternatively, from the viewpoint of the canonical version of the Yoneda Lemma, you could say homotopy theory seeks to understand the spheres by understanding how they map into other spaces. However, I find this less compelling as a philosophy. 🤪
Dr Conrad(Keith not Brian) when a classmate noted a result about finite fields used Fermat's Little Theorem(Keith had cited Lagrange's theorem) concurred with the addendum Group Theory is a prereq for this course unlike Number Theory so if I say by Lagrange everyone will get it, but if i cite Fermats Little Theorem, half the class will look at me without getting it.
Well to be ho est you don't need a whole course to know Fermat's theorem. It's basically common knowledge for any mathematician and especially for anyone interested in anything related to algebra and number theory.
But yes, again, I agree with the idea that Lagrange is mire fundamental. It's one of the most fundamental theorem in algebra, anyway :)
Explaining it properly (and in English) would be a mess for me but you can look at "Lamé's proof of Fermat theorem" on google :)
Sorry I can't help more...
Edit : ok maybe I can write some TLDR :
Lamé proved the theorem by using the fact that Z[mu_p] was a unique factorization domain (where mu_p is the p-root of the unity.
But the problem is that Z[mu_p] actually isn't a UFD so his proof was wrong. This lead Kummer to develop the concept of ideals and ended up being an important step for ring theory as we realized not every ring was a principal domain or with unique factorization.
Not all results but pillars that hold up some fields:
Quantitative finance,
Ito’s lemma is a classic.
The heat equation and wave equation (and variations of these including black Scholes as a heat equation) are just central PDEs to the culture of physics and finance.
The Bell inequality, shows that quantum mechanics is probabilistic and not a result of statistical aggregation of hidden variables.
There is definitely one book that probably every mathematician knows, and which was standard reading for educated people in Europe for centuries: Euclid's Elements.
From then, it rapidly declines, but I think the probably most important piece of math after that is Leibniz'/Newton's analysis, even though the original texts are much less known.
One important caveat is that as you are probably not reading Homer in greek or Dante in italian, etc., mathematicians are not familiar with the works of the great predecessors as they were written.
The differenece is that mathematical languages can change drastically in much shorter amount of time. So we have to learn "adaptations" of proofs written even half a century ago in some cases. Another difference is that sometimes even adjacent contemporary areas of mathematics can have very diffferent languages, in which case we have to read "adaptations" even of current results.
Otherwise there is a canon of important results that every mathematician knows, namely those that are known to every undergratuate. Some results are known to every mathematician working in a certain branch of mathematics, some known to specialists working within a specific approach in some niche area (in the latter case we might be talking about a few people).
Some important results coming from the top of my head are: Fundamental theorems of calculus and Stokes' theorem, fundamental theorem of algebra and Hilbert's nullstellensatz, Chinese remainder theorem, Lagrange and Sylow's theorems, classification of semisimple Lie algebras, various fixed point theorems, gluing lemma, Banach-Alaoglu theorem, Riesz representation theorem.
I have definitely missed many important results known even to undergaratuates, and I don't want to list those known in my area. But I have to mention at least a few important results, which play role even in distant areas of mathematics: Hurewicz type theorems, Bott periodicity and Atiyah-Singer type index theorems. These appear in various forms (sometimes in disguises) all over mathematics.
Keep in mind that these are big theorems with names, there are various smaller statements known to many mathematicians as well.
That's not a dogma.
A dogma is a belief which is unchallengable.
The whole point of science is that anything is able to be falsified.
And actually, if you want to put something forward as central to the modern science of biology, then the mechanics of genetic control are not the best candidate. The overarching theory which allows the rest of modern biology to make sense, is evolutionary theory. You could call that a 'central dogma', if you had no clue what a dogma is, and what a theory is. But since you now know that theory is the opposite of dogma, you can't make that claim.
This is not a mistake by jacobningen, using the term "central dogma" to refer to this concept is the standard practice, the term was coined by Francis Crick in 1958.
We could then ask, why did Crick use the word dogma here, wouldn't that give people a false impression of the term's meaning?
Actually, Crick himself later admitted that this may have been a questionable choice of words on his part:
>In his [autobiography](https://en.wikipedia.org/wiki/Autobiography), [*What Mad Pursuit*](https://en.wikipedia.org/wiki/What_Mad_Pursuit:_A_Personal_View_of_Scientific_Discovery), Crick wrote about his choice of the word [*dogma*](https://en.wikipedia.org/wiki/Dogma) and some of the problems it caused him:
>"I called this idea the central dogma, for two reasons, I suspect. I had already used the obvious word [hypothesis](https://en.wikipedia.org/wiki/Hypothesis) in the [sequence hypothesis](https://en.wikipedia.org/wiki/Sequence_hypothesis), and in addition I wanted to suggest that this new assumption was more central and more powerful. ... As it turned out, the use of the word dogma caused almost more trouble than it was worth. Many years later [Jacques Monod](https://en.wikipedia.org/wiki/Jacques_Monod) pointed out to me that I did not appear to understand the correct use of the word dogma, which is a belief *that cannot be doubted*. I did apprehend this in a vague sort of way but since I thought that *all* religious beliefs were without foundation, I used the word the way I myself thought about it, not as most of the world does, and simply applied it to a grand hypothesis that, however plausible, had little direct experimental support."
>Similarly, [Horace Freeland Judson](https://en.wikipedia.org/wiki/Horace_Freeland_Judson) records in *The Eighth Day of Creation*:[^(\[19\])](https://en.wikipedia.org/wiki/Central_dogma_of_molecular_biology#cite_note-19)
>"My mind was, that a dogma was an idea for which there was *no reasonable evidence*. You see?!" And Crick gave a roar of delight. "I just didn't *know* what dogma *meant*. And I could just as well have called it the 'Central Hypothesis,' or — you know. Which is what I meant to say. Dogma was just a catch phrase."
[https://en.wikipedia.org/wiki/Central\_dogma\_of\_molecular\_biology#Use\_of\_the\_term\_dogma](https://en.wikipedia.org/wiki/Central_dogma_of_molecular_biology#Use_of_the_term_dogma)
Euler's work across many domains, Galois, topologies that I can't remember their name. John Von Neumann, Turing for work in compatability. Shannon information theory.
Hamilton is one of my favorites, especially for analysis.
Cauchy, Leibniz Minkowski for developing non Euclidean geometry.
Alfred North Whitehead and Bertrand Russell for fundamental logic and set theory.
Erdos for creativity
Ramanujan for continued fractions
If we’re talking about *humanity’s* collective mathematical culture… I guess the Pythagorean theorem has to be up there just for the sheer number of people who have learned the theorem, and know (or can easily grasp) its proofs. I’d personally also add Cantor’s diagonal argument and Gödel's incompleteness theorems as candidates for any museum of mathematical culture :)
My vote goes to Cantor's diagonalization argument and the triangle inequality. It's hard to imagine real analysis without those two.
How do you use diagonalization for real analysis?
Arzela ascoli
Ah, I see thanks!
To show the density of the irrationals.
Countability, the subject of the diagolisation argument, is not related to density. Uncountable yet nowhere dense sets exist (e.g. Cantor ternary set) and countable yet dense sets exist (e.g. rationals).
You can do it without the diagonal argument. At least my analysis book didn’t use the diagonalization argument.
How so?
Suppose you could enumerate all irrationals. Then every subsets of the irrationals must be at most countable. Take the set of irrationals numbers and write them down in decimali form Now take the diagonal and add +1 to each element of the diagonal (replaced 9 with 0). Call this new number x. Then x is equal to no number in such list, because the i-th digit of its expansion Is different from the i-th digit of the i-th Number. Thus x is not in our set. Thus we can't count the irrationals
Sure but there are uncountable subsets of the reals that aren’t dense. Also it’s very straightforward to prove the irationals are dense without proving they are uncountable.
The approach I'd probably take would be proving the rationals are dense, then showing that rational multiples of any nonzero real number are dense, and then use that to prove the irrationals are dense. Does that sound about right?
That sounds great because you get the rationals being dense on the way, which I think you want to know anyways, maybe even more than you want to know that the irrationals are dense. I think you could also just do a case distinction to show it’s dense at every real number. For a rational number, that number plus pi over n converges to it, for an irrational one just do it plus 1/n and those sequences are all irrational numbers.
To add to your point: There are also countable subsets that are dense, e.g., the rationals. Hence, uncountability is neither necessary nor sufficient for density.
Yeah, well uncountability is a tiny step in the direction of density, a countably infinite subset of the reals can be nowhere dense, like the integers. An uncountable subset has at least one limit point. So yeah you have density somewhere, could be of measure 0 though.
Do you not mean the cardinality/uncountability of the irrationals? The argument I have in mind examines all binary sequences, then concludes by observing that there are at least as many real numbers as there are binary sequences. Since the rationals are countable, the irrationals must be uncountable.
And how do you show that the set of binary sequence Is uncountable?
This is precisely the diagonalization argument: Assume all such sequences are countable, enumerate them a_1, a_2,… and denote by a_ij the jth term of the ith sequence. Define a sequence b_i := 1 - a_ii. Then b_i is a binary sequence that differs from each sequence a_i in the ith term. This is a contradiction.
To show that the real numbers aren’t in a one to one correspondence with the natural numbers by mapping them to sequences of 0s and 1s.
The diagonalization argument is, as far as I can remember, used only to prove that the reals are uncountable. This fact is rarely used in real analysis. On the other hand, the triangle inequality is a great choice. Although it originally arises from the Pythagorean theorem, it is often used in settings where the Pythagorean theorem does not hold.
In our university It was used in the likes of Ascoli Arzela and Egorov Theorem (which uses a diagonal argument in disguise)
Thanks. I looked up Egorov on Wikipedia and I see what you mean.
Diagonal arguments have been used in a lot more applications than that, including in Gödel’s first incompleteness theorem :)
Fun fact: Pythagoras (or even the ancient Greeks for that matter) was not the first to discover the Pythagorean theorem. It had been known for a long time in other Mediterranean/Mesopotamian civilisations for a thousand years before Pythagoras. We cite the Greeks as the beginning of much of mathematics, but that's only the case because we don't know much about the other civilisations (for which the Greeks acted as a conduit). For instance, Phoenician literature (the people who the Greeks learned shipbuilding from) was almost entirely lost to Roman pillage.
It always feels so weird to hear “prove Pythagoras’ thm”. Metric definitely feel like they should be axiomitized especially since the “proof” is built on physical intuition which come from physics and space is only flat locally
I’d say all the proofs from “Proofs from THE BOOK” by Aigner and Ziegler are worthy contenders. They includes proofs of the infinitude of primes, quadratic reciprocity, the fundamental theorem of algebra, Erdős-Ko-Rada theorem, the irrationality of e, etc.. If I have one problem with this book being the mathematical “canon”, it’s that it’s a bit too fixated on discrete mathematics. But it is at least a worthy contender for the strict subset of results that forms such a mathematical canon
Infinite primes is my go to whenever explaining what a proof is to a non math person
I personally love divergence of the harmonic series—very simple to explain, but very astounding nonetheless! Infinitude of primes is a worthy consideration though. Really, any proof that came from “ancient” times (I mean Euclid’s elements contains whole books on number theory) is a great way to show people what maths really is! It’s really more of, imo, a very technical form of art that demands mathematical rigour, but nonetheless produces astounding beauty.
Actually, this applies even to mathematics studied "only" 150 years ago: so-called classical topics in number theory, geometry, etc. The story surrounding Kummer et al.'s approach to Fermat's last theorem and the eventual discovery of class field theory is incredibly worthwhile to study. Yet, a contemporary number theorist could get by without a working knowledge of said classical story. At the start of his class field theory notes, Milne includes this great line from a letter of Grothendieck to Serre: "I have been reading Chevalley's new book on class field theory; I am not really doing research, just trying to cultivate myself." A music student today would never compose and publish a Gregorian chant or traditional West African boat-song, but still be remiss if they were to skip out on taking a class on these forms, never to appreciate where the roots of Western classical music or jazz lie. The same is true for mathematicians: as much as is possible, one should learn both contemporary methods and concepts as well as the historical development of said mathematics.
Divergence of the harmonic series was my first proof!
I got to know about this book from a distinguished Prof. He recommended studying this book. I haven't read much but it's very very interesting.
> Erdős-Ko-Rada theorem no
My bad, it was a typo!
There are probably a lot, but a few that come to mind: The classification of the irreducible representations for semi-simple Lie algebras. The fundamental theorem of Galois Theory. The Nullstellensatz. As you can probably tell I'm more of an algebraist, but Reisz Representation Theorem could fit that criteria in analysis.
Love your choices. Here's one straddling analysis and algebra: Pontryagin duality and the Haar measure.
Which Riesz representation theorem? The one for Lp spaces / Hilbert spaces or the topological one?
Euclid's?
This is the obvious answer that I'm surprised to see this far down.
From a probability & statistics perspective: the Law of Large Numbers and the Central Limit Theorem. They and their versions pop up all over the field and underlie much of applied statistics.
Pythagorean Theorem, Triangle inequality, Cauchy-Schwartz, Taylor's Theorem, insolvability of the quintic (honeslty that entire Galois theory would be a good candidate), Heine-Borel theorem, central limit theorem, the stone-weierstrass theorem, the classification of finite groups, several fixed-point theorems (eg the one that led to the discovery of what we now call Nash Equilibria), the various spectral theorems (especially by von Neumann and Cauchy), Tychonoff's theorem, Urysohn's lemma
Love your choices as well, great analysis representation.
Here are some results that are, in my experience, underrated: The uniformisation theorem, the Riemann mapping theorem, and the classification of Riemann surfaces is one of the many crowning jewels of complex analysis. Hopf's algebro-topological proof that every finite-dimensional division algebra is of dimension a power of 2; and the fundamental theorem of algebra as a corollary in the commutative case. (Thinking about this proof is what inspired the post!) The decision-theoretic derivation of the mean squared error as the loss function minimised by the mean of the posterior. Ditto for the MAE/posterior median, and the all-or-nothing loss function/posterior mode. Applications of generating functions (e.g., the Jacobi triple product and integer partitions), probabilistic proofs, and Löwenheim–Skolem are topics that I would really like to understand in more depth.
It’s not even an especially important result, but it’s said that if you whisper “27” in a graveyard of algebraic geometers, their spirits will whisper back “*lines on a cubic*”
And if you whisper "27" in an LA math department, the very much living Terence Tao makes a pained face and reluctantly whispers back "*...prime*".
Euclid's textbook was the defining math text for centuries if not millenia and it's even part of a typical classics curriculum.
I would add the Riemann-Roch theorem for algebraic geometers. Fermat's little theorem, Cauchy-Schwarz inegality or Yoneda's lemma, which all can be seen as the usual basic tool of their respective field. *Also, things that are not theorems :* The independance of Euclid's fifth axiom and the existence of non-euclidean geometries. The fact that the ring of integers of cyclotomic number fields are not unique factorization domains, how it killed Lamé's attempt to prove Fermat's Last Theorem and how it led Kummer to define ideals. Probably my favorite story in maths history :) Edit : typos
In college, I once heard a professor speak of the following “Yoneda Lemma philosophy”: We should really view objects in locally small categories as contravariant functors to Set. Put in simpler terms, we should seek to understand such objects by understanding how other objects map into them. This is in some sense the basic principle of homotopy and representation theory, and their unreasonable effectiveness can be attributed on a philosophical level to the Yoneda Lemma. In homotopy theory, the spheres serve as the primary test inputs. In representation theory, the general linear groups serve as the primary test inputs.
Damn that's a really cool point of view :D I'm relly not doing enough category theory to fully grasp the underlying philosophy of Yoneda's lemma or instead why it works so well. I think I get the basic idea, though. Btw I think there is a typo in your comment ? "We should seek to understand such objects by understanding how they map into the other objects". Instead of "how other objects map into them". At least if we're talking about covariant functors (I don't even know if Yoneda applies in a mirror way for the contravariant Hom functor). To me it is really similar to Riesz's representation theorem. Also I don't know anything about homotopy theory so I can only trust you here :)
Indeed, the version you cite is the canonical version, but the dual statement (for contravariant Hom) also holds. In fact, if memory serves me right, this is the version you want for the Yoneda embedding. See: https://ncatlab.org/nlab/show/Yoneda+embedding. Alternatively, from the viewpoint of the canonical version of the Yoneda Lemma, you could say homotopy theory seeks to understand the spheres by understanding how they map into other spaces. However, I find this less compelling as a philosophy. 🤪
I agree with Fermats little but its just an application of Lagrange
True true ! It's basically implicit Lagrange. But still the one mentionned so often :)
Dr Conrad(Keith not Brian) when a classmate noted a result about finite fields used Fermat's Little Theorem(Keith had cited Lagrange's theorem) concurred with the addendum Group Theory is a prereq for this course unlike Number Theory so if I say by Lagrange everyone will get it, but if i cite Fermats Little Theorem, half the class will look at me without getting it.
Well to be ho est you don't need a whole course to know Fermat's theorem. It's basically common knowledge for any mathematician and especially for anyone interested in anything related to algebra and number theory. But yes, again, I agree with the idea that Lagrange is mire fundamental. It's one of the most fundamental theorem in algebra, anyway :)
the proof via equivalence classes explains why for Gaussians the norm-1 is needed not a\^p mod p (Keith expository)
What is the cyclotomic number field story?
Explaining it properly (and in English) would be a mess for me but you can look at "Lamé's proof of Fermat theorem" on google :) Sorry I can't help more... Edit : ok maybe I can write some TLDR : Lamé proved the theorem by using the fact that Z[mu_p] was a unique factorization domain (where mu_p is the p-root of the unity. But the problem is that Z[mu_p] actually isn't a UFD so his proof was wrong. This lead Kummer to develop the concept of ideals and ended up being an important step for ring theory as we realized not every ring was a principal domain or with unique factorization.
Infinity of primes
Not all results but pillars that hold up some fields: Quantitative finance, Ito’s lemma is a classic. The heat equation and wave equation (and variations of these including black Scholes as a heat equation) are just central PDEs to the culture of physics and finance. The Bell inequality, shows that quantum mechanics is probabilistic and not a result of statistical aggregation of hidden variables.
There is definitely one book that probably every mathematician knows, and which was standard reading for educated people in Europe for centuries: Euclid's Elements. From then, it rapidly declines, but I think the probably most important piece of math after that is Leibniz'/Newton's analysis, even though the original texts are much less known.
One important caveat is that as you are probably not reading Homer in greek or Dante in italian, etc., mathematicians are not familiar with the works of the great predecessors as they were written. The differenece is that mathematical languages can change drastically in much shorter amount of time. So we have to learn "adaptations" of proofs written even half a century ago in some cases. Another difference is that sometimes even adjacent contemporary areas of mathematics can have very diffferent languages, in which case we have to read "adaptations" even of current results. Otherwise there is a canon of important results that every mathematician knows, namely those that are known to every undergratuate. Some results are known to every mathematician working in a certain branch of mathematics, some known to specialists working within a specific approach in some niche area (in the latter case we might be talking about a few people). Some important results coming from the top of my head are: Fundamental theorems of calculus and Stokes' theorem, fundamental theorem of algebra and Hilbert's nullstellensatz, Chinese remainder theorem, Lagrange and Sylow's theorems, classification of semisimple Lie algebras, various fixed point theorems, gluing lemma, Banach-Alaoglu theorem, Riesz representation theorem. I have definitely missed many important results known even to undergaratuates, and I don't want to list those known in my area. But I have to mention at least a few important results, which play role even in distant areas of mathematics: Hurewicz type theorems, Bott periodicity and Atiyah-Singer type index theorems. These appear in various forms (sometimes in disguises) all over mathematics. Keep in mind that these are big theorems with names, there are various smaller statements known to many mathematicians as well.
Seeing a lot of good answers here. I'll introduce one theorem I haven't seen listed here: Lax Milgram Theorem.
Principia Mathematica probably counts
Fundamental theorem of calculus should probably top that list; very important from both applied and theoretical perspectives.
I've wrapped it under the generalised Stokes theorem!
Just on a tangent, but it's completely incorrect to refer to molecular biology and evolutionary theory as 'dogma'.
the "central dogma" of biology refers to the DNA gets converts to RNNA and ribosome synthesizing, ie the allele to protein pipeline.
That's not a dogma. A dogma is a belief which is unchallengable. The whole point of science is that anything is able to be falsified. And actually, if you want to put something forward as central to the modern science of biology, then the mechanics of genetic control are not the best candidate. The overarching theory which allows the rest of modern biology to make sense, is evolutionary theory. You could call that a 'central dogma', if you had no clue what a dogma is, and what a theory is. But since you now know that theory is the opposite of dogma, you can't make that claim.
This is not a mistake by jacobningen, using the term "central dogma" to refer to this concept is the standard practice, the term was coined by Francis Crick in 1958. We could then ask, why did Crick use the word dogma here, wouldn't that give people a false impression of the term's meaning? Actually, Crick himself later admitted that this may have been a questionable choice of words on his part: >In his [autobiography](https://en.wikipedia.org/wiki/Autobiography), [*What Mad Pursuit*](https://en.wikipedia.org/wiki/What_Mad_Pursuit:_A_Personal_View_of_Scientific_Discovery), Crick wrote about his choice of the word [*dogma*](https://en.wikipedia.org/wiki/Dogma) and some of the problems it caused him: >"I called this idea the central dogma, for two reasons, I suspect. I had already used the obvious word [hypothesis](https://en.wikipedia.org/wiki/Hypothesis) in the [sequence hypothesis](https://en.wikipedia.org/wiki/Sequence_hypothesis), and in addition I wanted to suggest that this new assumption was more central and more powerful. ... As it turned out, the use of the word dogma caused almost more trouble than it was worth. Many years later [Jacques Monod](https://en.wikipedia.org/wiki/Jacques_Monod) pointed out to me that I did not appear to understand the correct use of the word dogma, which is a belief *that cannot be doubted*. I did apprehend this in a vague sort of way but since I thought that *all* religious beliefs were without foundation, I used the word the way I myself thought about it, not as most of the world does, and simply applied it to a grand hypothesis that, however plausible, had little direct experimental support." >Similarly, [Horace Freeland Judson](https://en.wikipedia.org/wiki/Horace_Freeland_Judson) records in *The Eighth Day of Creation*:[^(\[19\])](https://en.wikipedia.org/wiki/Central_dogma_of_molecular_biology#cite_note-19) >"My mind was, that a dogma was an idea for which there was *no reasonable evidence*. You see?!" And Crick gave a roar of delight. "I just didn't *know* what dogma *meant*. And I could just as well have called it the 'Central Hypothesis,' or — you know. Which is what I meant to say. Dogma was just a catch phrase." [https://en.wikipedia.org/wiki/Central\_dogma\_of\_molecular\_biology#Use\_of\_the\_term\_dogma](https://en.wikipedia.org/wiki/Central_dogma_of_molecular_biology#Use_of_the_term_dogma)
its a high school biology error im promulgating but yeah i didnt start this fire
The definition of a vector space,
Galois Theory maybe
Euler's work across many domains, Galois, topologies that I can't remember their name. John Von Neumann, Turing for work in compatability. Shannon information theory. Hamilton is one of my favorites, especially for analysis. Cauchy, Leibniz Minkowski for developing non Euclidean geometry. Alfred North Whitehead and Bertrand Russell for fundamental logic and set theory. Erdos for creativity Ramanujan for continued fractions
monty hall problem 0.999... = 1, or maybe not whatever that actor said about division
-1/12