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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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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
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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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78 Answers

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
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Computer science: sort requires n * log n

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Yijie Han dx.doi.org/10.1016/j.jalgor.2003.09.001 has an algorithm for sorting natural numbers that takes time O(n log log n). So it really depends on the class of sorting algorithms. Any comparison sort needs at least $\log_2 n! > c n \log n$ comparisons. –  Konrad Swanepoel Nov 11 '09 at 10:09
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Algebraic number theory: Hilbert 90.

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Algebraic Geometry: Bezout's Theorem. It's also good for selling what algebraic geometry is to people who've never heard of it before.

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Bezout's Theorem is a nice theorem but it is hardly surprising in its proper setting (algebraically closed field, taking into account multiplicities and points at infinity). –  lhf Oct 30 '09 at 9:18
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But it is a great motivator for schemes and cohomology: To put projective and affine space in the same framework, you need gluing. To get the right formulas for higher order contact, you need the scheme theoretic intersection of curves. When you approach the theorem cohomologically, it reduces to just intersecting lines (which is a conceptually beautiful way to approach the proof). So not only is it simple to understand, it can be used as motivation for very deep ideas. –  Steven Gubkin Nov 12 '09 at 20:04
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Graph theory: necessary and sufficient conditions for the existence of an Eulerian walk/cycle

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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
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Algebraic topology: Fundamental group of S^1.

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I'm not sure I like this one - although it can be hard to prove when you don't have much machinery, it's basically a tautology deriving from the definition of the fundamental group and not a very insightful result. I'd say the first nontrivial theorem of algebraic topology (at least on the homology side of things) is Poincaré duality - after all that was one of the things that launched the subject. Although there are easier results (Brouwer fixed point), they do not really pertain to algebraic topology itself. Considering homotopy, there should be something about homotopy groups of spheres. –  Sam Derbyshire Oct 25 '09 at 2:47
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This is an important first result because now we can prove the Brouwer fixed point theorem (usually the fundamental group comes before homology right? the other things you need for that result are much more straightforward). –  Sean Tilson Mar 5 '10 at 21:05
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This is the most important computation in algebraic topology, in my opinion. Everything else ultimately derives from it. –  Jeff Strom Jul 23 '10 at 18:06
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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 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
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Category theory: The Yoneda lemma.

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I thought about that, but I think category theorists would consider the Yoneda lemma trivial. Not to say that it's easy to understand, but it does follow directly from the category axioms. –  Qiaochu Yuan Oct 24 '09 at 20:36
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Though Yoneda's lemma isn't non-trivial, I feel like understanding its significance is definitely a non-trivial step. Schur's lemma in representation theory and Nakayama's lemma in algebra have a similar feel to them. They're pretty trivial to prove, but can take a while to really grok them. –  Anton Geraschenko Oct 24 '09 at 20:56
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Maybe it's more correct to say that Yoneda is the last trivial theorem? –  Harrison Brown Oct 25 '09 at 5:52
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Finite group theory: I would say Lagrange's theorem, that the order of a subgroup divides the order of the group. Certainly it's prior to the Sylow theorems, certainly it has content.

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I was about to post that. I should make the addendum though that while Lagrange stated his theorem in 1770, the first full proof occurred 30 years later at the same time as the first proof of the insolubility of the quintic. –  Jason Dyer Oct 24 '09 at 20:42
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LaGrange's theorem is just an application of the definition of an equivalence class. –  Harry Gindi Dec 13 '09 at 15:26
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Fermat's last theorem is just an application of the definition of a natural number. –  Steven Gubkin Mar 5 '10 at 0:15
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@Marcos : "I don't understand why he didn't prove it generally". Presumably because the definition of a group was only given in the mid-19th century. –  Laurent Berger Mar 1 '12 at 14:03
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Number theory: Different undergraduate textbooks approach the subject differently, of course. But the irrationality of the square root of 2 and the infinitude of primes are contentful theorems that are certainly very early historically, and also very early in at least some textbook treatments of the subject.

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I have to admit when I say "number theory" I always think "in the sense of Gauss." The infinitude of the primes is a better candidate now that I think about it. –  Qiaochu Yuan Oct 24 '09 at 20:39
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Perhaps you could even argue that the irrationality of the square root of 2 and the infinitude of primes are the first nontrivial theorems in mathematics as a whole. The Pythagorean theorem seems to be another candidate in this direction. –  Michael Lugo Oct 24 '09 at 21:59
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Probably Pythagorean before "sqrt(2) is irrational". Without a^2 + b^2 = c^2 with a = b = 1, we have no reason to consider sqrt(2) in the first place. –  Chad Groft Feb 22 '10 at 17:31
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Complex analysis: Riemann mapping theorem.

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

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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
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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
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Quadratic reciprocity feels to me like a "first nontrivial statement" without an obvious branch of mathematics. Not number theory -- there are things like the infinitude of the primes, and "algebraic number theory" doesn't seem quite right either...

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The first nontrivial statement in class field theory? –  James Cranch Mar 1 '12 at 15:35
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Complex analysis: Hadamard's factorization theorem.

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

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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
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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
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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
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Combinatorics: the nth Catalan number is (2n choose n)/(n+1)

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Linear algebra: rank-nullity theorem.

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Although of course, this is just a categorification of the corresponding statement for finite sets... –  Scott Morrison Oct 24 '09 at 21:42
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You could say that the rank nullity theorem is a direct consequence of the fact that all modules over fields are free (and hence projective). This may be a warped way of looking at things but I have to admit, thats how I remember it now :) –  Grétar Amazeen Oct 25 '09 at 2:08
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I think that various theorems about bases (every two bases have the same cardinality, every linearly independent set can be extended to a basis, and so on) are nontrivial and certainly come earlier. Why non-trivial? Because they may fail for modules over other rings, even if the module is free (but the ring may be noncommutative) or for f.g. modules over commutative rings (maximal linearly independent systems may have different cardinality, mathoverflow.net/questions/30066/…) –  Victor Protsak Jul 24 '10 at 5:33
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Ring theory: If R is a UFD, then so is R[x].

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For ring theory, I prefer R noetherian implies R[x] is. The Hilbert Basis Theorem. –  Charles Siegel Oct 26 '09 at 20:56
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Post this as a separate answer so that it can get voted independently. –  lhf Oct 30 '09 at 9:20
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Functional Analysis: the Hahn-Banach Theorem.

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Probability theory: the Central Limit Theorem?

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I would consider the law of large numbers non-trivial, and that was covered in my probability courses long before the CLT. –  Kevin P. Costello Oct 25 '09 at 0:19
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The uncountability of the reals. This is such a classic fact that I'm not sure how to classify it. Set theory, perhaps.

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Theory of Computation: The halting problem

Set Theory: Cardinality of a set is strictly less than its powerset

Not sure which field (one might learn it in an intro real analysis course): Reals are uncountable

Complexity theory: The Time and Space Hierarchy theorems

One might learn these results in different courses, but they're all the same beautiful idea: Cantor's diagonalization.

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Differential equations: Picard's theorem on existence and uniqueness.

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Algebraic topology: Poincaré Duality

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Knot theory: sufficiency of the Reidemeister moves.

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operator algebras: the Gelfand transform is an isomorphism for commutative $C^*$-algebras.

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In mathematical logic, Gödel's incompleteness theorem.

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I would think of Completeness or Compactness of first order logic as a more reasonable answer. Incompleteness is interesting, but it feels more like a side branch of the theorem tree than the main trunk (whatever that means). –  Richard Dore Oct 25 '09 at 6:40
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Functional analysis: the open mapping theorem.

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Homotopy Theory: the Hopf Fibration?

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That's not a theorem. –  André Henriques Jul 23 '10 at 17:44
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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
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Symplectic Geometry: Darboux's theorem

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