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In most students' introduction to rigorous proof-based mathematics, many of the initial exercises and theorems are just a test of a student's understanding of how to work with the axioms and unpack various definitions, i.e. they don't really say anything interesting about mathematics "in the wild." In various subjects, what would you consider to be the first theorem (say, in the usual presentation in a standard undergraduate textbook) with actual content?

Some possible examples are below. Feel free to either add them or disagree, but as usual, keep your answers to one suggestion per post.

  • Number theory: the existence of primitive roots.
  • Set theory: the Cantor-Bernstein-Schroeder theorem.
  • Group theory: the Sylow theorems.
  • Real analysis: the Heine-Borel theorem.
  • Topology: Urysohn's lemma.

Edit: I seem to have accidentally created the tag "soft-questions." Can we delete tags?

Edit #2: In a comment, ilya asked "You want the first result after all the basic tools have been introduced?" That's more or less my question. I guess part of what I'm looking for is the first result that justifies the introduction of all the basic tools in the first place.

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closed as no longer relevant by Felipe Voloch, Suvrit, Bill Johnson, Todd Trimble, Qiaochu Yuan Mar 1 '12 at 16:41

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I wish the system was smart enough to understand that example = examples and soft-question = soft-questions (and topos = topoi :) ). – Ilya Nikokoshev Oct 24 '09 at 20:34
@Qiaochu: If there is no question with a given tag, the tag will disappear on its own within about a day. If you find two tags that should really be the same, flag the post for moderator attention with a comment explaining that the tags should be merged. @ilya: the site doesn't understand the meanings of the tags, so it would probably cause trouble if it tried to automatically merge tags. But you usually don't write the complete name of the tag anyway; you just type the first few letters and then you're shown a list of tags that match. – Anton Geraschenko Oct 24 '09 at 20:50
@Anton, the object = objects idea seem to be easy to implement to me -- is there an example where this merging makes no sense? – Ilya Nikokoshev Oct 24 '09 at 23:20
Is anyone other than me in favor of changing the word "non-trivial" to "interesting" in the title? I, for one, would feel much less intimidated. – S. Carnahan Oct 26 '09 at 9:45
This question seems to have long outgrown its usefulness (and some of the more recent additions have been, IMHO, lousy). Voting to close. – Todd Trimble Mar 1 '12 at 16:03

77 Answers 77

Additive combinatorics: Roth's theorem (that a dense subset of $\{1,2,...,N\}$ contains an arithmetic progression of length 3). It's extraordinary how much of the subject opens up once one has seen just this theorem proved, and it can be done quite easily from first principles.

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Now, isn't Cauchy-Davenport already quite interesting? – darij grinberg May 23 '10 at 22:05

Finite geometry: The Bruck-Ryser-Chowla Theorem. If a finite projective plane of order q exists and q is congruent to 1 or 2 (mod 4), then q must be the sum of two squares.
BRC also has the distinction of being the la(te)st non-trivial theorem in finte geometry/design theory, as it's been the strongest result on existence of projective planes/symmetric designs for a given class of orders q for the past 60 years.

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Excluding, of course, the result of Lam, Thiel and Swiercz on the plane of order 10. In combinatorial design theory more generally, there are certainly other non-trivial results. – Will Orrick Oct 25 '09 at 0:00
I agree. I was trying to keep the post short and did not want to put in too much. Perhaps a minor edit is in order. – Sonia Balagopalan Oct 25 '09 at 0:25
I remember thinking this was a really beautiful theorem when I saw it as an undergrad. I haven't thought about it for a while, thanks for reminding me :) – Grétar Amazeen Oct 25 '09 at 2:00

Homotopy Theory: the Hopf Fibration?

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That's not a theorem. – André Henriques Jul 23 '10 at 17:44
There is a theorem there (more than one, in fact), e.g. "$\pi_3(S^2)$ is an infinite cyclic group generated by the class of the Hopf fibration". – Victor Protsak Jul 24 '10 at 5:37

Differential Geometry (of surfaces, say): the Gauss-Bonnet theorem.

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Certainly non-trivial, but comes after Theorema Egregium (oops... I've already left an equivalent comment at Theorem Egregium entry). – Victor Protsak Jul 24 '10 at 5:54

Points on an elliptic curve form an abelian group, accredited to Fermat.

This should appear as an example when one introduces group theory. One can easily state many non-trivial facts, like the rational points form a finitely generated subgroup, whose torsion is known (Mazur), and whose rank is the subject of Birch-Swinnerton-Dyer conjecture. (That would also be a nice example of the classification theorem of finitely generated abelian groups)

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I'd call this an example in group theory but a theorem in algebraic geometry. – Qiaochu Yuan Oct 27 '09 at 2:38
I actually did attend a short class about the classification theorem for finite abelian groups which discussed Mordell's Theorem on the last day. – David Corwin May 9 '10 at 15:22
How about nominating Mordell's theorem (finite generation of $E(\mathbb{Q})$ for the first non-trivial theorem in arithmetic algebraic geometry instead? – Victor Protsak Jul 24 '10 at 5:40

Poset theory: Dilworth's theorem.

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In geometric probability: Buffon's noodle (and needle).

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In analytic number theory: Euler's proof of the infinitude of primes, using the divergence of the harmonic series.

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So, not Euclid's proof, then? – Ryan Reich Jul 23 '10 at 22:23
See the word in italics. :) – David Hansen Jul 24 '10 at 20:45
Did it not used to say Euclid? I can't believe I made that mistake :) – Ryan Reich Jul 24 '10 at 21:28

Math/Number Theory: $\mathbb{Z}$ is an Euclidean domain..PID .. UFD

Linear Algebra: Every vector space has a basis and every two basis have the same cardinal.

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Harmonic Analysis: Plancherel's theorem

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The first interesting theorem in Differential Algebra is...

Liouville´s condition for integration of elementary functions in finite terms.

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Algebraic number theory: Hilbert 90.

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Interesting. Do you consider "rings of integers are Dedekind domains" to be trivial? – Qiaochu Yuan Oct 24 '09 at 20:25
No, certainly not trivial, though simple, and a good answer to the original question indeed. – Ilya Nikokoshev Oct 24 '09 at 20:36
On the other hand, the same applies to your examples of topology and group theory imho. – Ilya Nikokoshev Oct 24 '09 at 20:37

Complex analysis: Riemann mapping theorem.

(Easier candidates include: Liouville's theorem, Cauchy's integral formula, Picard theorems.)

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I didn't see your answer before I posted mine (Hadamard's theorem). I guess Hadamard's factorization theorem and Riemann's mapping theorem are pretty much independent, so either one could potentially be the "first nontrivial theorem." :-) – Darsh Ranjan Oct 24 '09 at 20:59
At least for me, taking this course as an undergraduate, it was the Cauchy integral formula for sure. – JSE Oct 24 '09 at 22:36
he Cauchy integral formula may be under-appreciated these days because we've moved it up early in the curriculum. We think it's elementary because it's presented early on. But historically it came after much of the material now in a complex analysis course. It was moved up precisely because it is so powerful and makes the proofs of other theorems easier. – John D. Cook Oct 26 '09 at 4:39

The representation theory of compact groups: The Peter-Weyl Theorem.

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Lie Algebras: Simple Lie algebras can be recovered from their Dynkin diagrams via Serre relations. (Maybe you can argue that PBW is really the first non-trivial fact)

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I would suggest Weyl complete reducibility. – Steven Sam Oct 24 '09 at 22:24
I agree it's non-trivial if you try to prove it purely algebraically, but the proof using the unitary trick on the associated simply connected compact group is pretty easy to digest. – Dinakar Muthiah Oct 25 '09 at 5:09
The construction of a compact form requires nontrivial amounts of structure theory. In fact, I conjecture that most people who know about unitary trick have never seen a complete proof. Fortunately, algebraic arguments based on Casimir allow one to bypass these complications. – Victor Protsak Jul 24 '10 at 5:45

Commutative algebra: primary decomposition.

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Really? Atiyah-Macdonald, at least, doesn't seem to care about primary decomposition. – Qiaochu Yuan Oct 25 '09 at 21:25
It's chapter 4 in A-M according to Google books; the previous chapters are on rings, modules, and localization, and are (if I remember correctly, or guess correctly based on the topic list) basically definition-oriented. – Akhil Mathew Oct 26 '09 at 1:04
I may have misunderstood the introduction. The beginning of the chapter: "In the modern treatment, with its emphasis on localization, primary decomposition is no longer such a central tool in the theory. It is still, however, of interest in itself and in this chapter we establish the classical uniqueness theorems." – Qiaochu Yuan Oct 26 '09 at 17:04
I'd argue for Nakayama on the same grounds as the Yoneda lemma, but again that's of a rather different nature. – Harrison Brown Nov 27 '09 at 4:07

Analysis/Topology. The closed interval [a,b] is compact.

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Not with the metric definition "compact" = "closed and bounded", because then it's obvious. Perhaps you mean the Heine–Borel theorem, but that was already listed in the original question. – lhf Oct 30 '09 at 9:13
lhf, the unit ball of an infinite-dimensional Hilbert space is closed and bounded, but not compact. – Todd Trimble Mar 1 '12 at 13:29

Two fields, one first nontrivial theorem:

Graph theory: Hall's marriage theorem.

Majorization theory: Birkhoff's theorem that the set of all doubly symmetric matrices is the convex hull of the permutation matrices.

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Algebra: Classification of finite abelian groups

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Combinatorics: counting the number of derangements of [n].

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I'm not at all convinced that this is the best answer, but after a few minutes of scratching my head, I turned to an old undergraduate textbook. Six chapters in, this is the first proven result that seems to qualify. – Jonah Ostroff Oct 24 '09 at 21:08
I think I would disagree, since counting derangements is just a standard application of inclusion-exclusion, which is pretty trivial. – Harrison Brown Oct 24 '09 at 21:30
Combinatorics doesn't really fall under the purvey of this question, since it's both relatively non-axiomatic and highly non-linear. And for what it's worth, I consider inclusion-exclusion highly nontrivial, at least conceptually (as Mobius inversion). – Qiaochu Yuan Oct 24 '09 at 21:32
I think this answer is reasonable. But this implicitly requires that permutations be one of the first objects you look at. As Qiaochu has pointed out, we don't necessarily have to make this choice. – Michael Lugo Oct 24 '09 at 21:56

Banach algebras: Gelfand's proof of the Wiener lemma for l1(Z)

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(I suppose someone will point out that this rests on the Gelfand-Mazur theorem, but I maintain that Wiener's lemma is more arresting and grabs the attention better - hence is more "interesting" in that sense. Obviously the Gelfand-Mazur theorem lies deeper and is in that sense more "interesting") – Yemon Choi Nov 11 '09 at 10:16

Nonlinear programming/Optimization: The Karush-Kuhn-Tucker conditions

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In real analysis, I would say The intermediate value theorem.

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Measure theory: the Hahn decomposition theorem.

If one were to attempt to simply union together all positive sets, one may end up with an uncountable union, which is thus not necessarily measurable. The fact that you can decompose the space into a positive and negative set is therefore a little surprising. The constructions in the proof of this theorem are typically delicate.

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Homological algebra: the cup product in (co)homology is graded commutative.

Is there a good reference for proofs of this in different cohomological theories? I know two proofs in simplicial homology and one proof in Hochschild cohomology... but I am far from seeing the relation between all these cup products.

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Do you mean that everything before cup product is trivial? Really? – Victor Protsak Jul 24 '10 at 5:49
The question is about "interesting", not about "nontrivial". "Interesting", for me, means a theorem which gets us more than we could have expected by looking at the condition and thinking about it for a while. I don't think the long exact sequences, the homotopic equivalence of resolutions and the basic properties of cup (not the commutativity) would fall under this, even if they are certainly not easy to come up with from scratch. – darij grinberg Jul 24 '10 at 15:02
You'll find in the arXiv a little note of mine which proves the commutativity of cup products in lots of cohomology theories. – Mariano Suárez-Alvarez Jul 24 '10 at 21:10
To be precise, the question asks for "the first theorem... with the actual content". Long exact sequence and homotopic equivalence of resolutions may not pass your muster, but what about, say, Hilbert's syzygy theorem? There are whole books on homological algebra that do not touch cup products(Gelfand-Manin is one, if I remember correctly). – Victor Protsak Jul 25 '10 at 5:54
Mariano: I know, but I don't know derived categories (still). – darij grinberg Jul 25 '10 at 11:40

Theoretical computer science or combinatorics or algorithmics: The diamond lemma.

Some days ago, while giving a seminar talk about Clifford algebras, I realized that Lawson-Michelson has a flawed proof that the canonical inclusion of a vector space in its own Clifford algebra is indeed injective (unfortunately, not until I had written this proof on desk). Most other literature gives ugly proofs using orthogonalization. Fact is, this injectivity works in a much more general context (namely, it works for any module over a commutative ring with $1$), where of course there needs not be any orthogonalization. And it is easily proven using the diamond lemma. A similar assertion for Weyl algebras is also clear from the diamond lemma, and so is the Poincaré-Birkhoff-Witt theorem (which is proven in intricate and opaque ways in most of literature). Maybe the problem is that geometers don't know enough computer science?

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Game Theory:

A zero-sum 2x2 (two person) matrix game which has no dominating strategy has an optimal mixed strategy, and the game is fair (0 expected value) if the determinant of the payoff matrix (say from the row player's point of view) is zero.

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Real Analysis: The function $t \mapsto \exp(it)$, defined by a certain power series, is periodic with a period of about 6.4.

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The second theorem is that 6.3 is closer. – Andreas Blass Mar 1 '12 at 14:43

Algebraic geometry: points are prime ideals.

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I would call this more a modification of the definition of "point" than anything else. But I would agree that something like "the Nullstellensatz" is a great answer here. – Qiaochu Yuan Oct 24 '09 at 20:09
I feel like non-trivial fact 1 is that "points are maximal ideals", and then non-trivial fact 2 is that "oh wait, points should really be prime ideals", followed by contemplating the nature of what a point is. – Greg Muller Oct 24 '09 at 20:10
Mm, depends on your taste certainly. But, ok, let's define the point geometrically as the minimal subscheme then it's a theorem. – Ilya Nikokoshev Oct 24 '09 at 20:12
I feel like history tells us that "points are maximal ideals" is non-trivial fact 1, then perhaps the Nullstellensatz is fact 2, and then... and then "oh wait, points are really prime ideals" is non-trivial fact \omega. Followed by \omega + 1, and so on. But of course history is often wrong. – Harrison Brown Oct 24 '09 at 20:26
I feel like non-trivial fact 1 is that "points are maximal ideals" <== this is exactly Euclid's "a point is what has no parts". Though I'm not quite saying this makes it trivial... – darij grinberg May 23 '10 at 22:11

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