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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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  • $\begingroup$ I wish the system was smart enough to understand that example = examples and soft-question = soft-questions (and topos = topoi :) ). $\endgroup$
    – Ilya Nikokoshev
    Commented Oct 24, 2009 at 20:34
  • $\begingroup$ @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. $\endgroup$
    – Anton Geraschenko
    Commented Oct 24, 2009 at 20:50
  • $\begingroup$ @Anton, the object = objects idea seem to be easy to implement to me -- is there an example where this merging makes no sense? $\endgroup$
    – Ilya Nikokoshev
    Commented Oct 24, 2009 at 23:20
  • $\begingroup$ 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. $\endgroup$
    – S. Carnahan ♦
    Commented Oct 26, 2009 at 9:45
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    $\begingroup$ This question seems to have long outgrown its usefulness (and some of the more recent additions have been, IMHO, lousy). Voting to close. $\endgroup$
    – Todd Trimble
    Commented Mar 1, 2012 at 16:03

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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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    $\begingroup$ 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. $\endgroup$
    – Jason Dyer
    Commented Oct 24, 2009 at 20:42
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    $\begingroup$ LaGrange's theorem is just an application of the definition of an equivalence class. $\endgroup$
    – Harry Gindi
    Commented Dec 13, 2009 at 15:26
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    $\begingroup$ Fermat's last theorem is just an application of the definition of a natural number. $\endgroup$
    – Steven Gubkin
    Commented Mar 5, 2010 at 0:15
  • $\begingroup$ It is even more trival in the case in which he stated it: If a polynomial has it's n variables permuted obtaining n! formally different expressions and by manipulating it is found out that they allot to only k different polynomials, then k divides n!. Proof: being all the n! expressions essentially equal, they must all group up in sets of d expressions that yield the same polynomial by manipulation. Hence k=n!/d. I don't understand why he didn't prove it generally. Did he state it generally? Did he need it? Has someone here read Lagrange's work? $\endgroup$
    – Marcos Cossarini
    Commented Mar 30, 2010 at 5:17
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    $\begingroup$ @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. $\endgroup$
    – Laurent Berger
    Commented Mar 1, 2012 at 14:03
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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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  • $\begingroup$ This was due to Cantor, right? My understanding is that he was working on perfect sets when thinking about the "cardinality" of the reals (Was it even referred to as cardinality at the time?). Although the word "sets" is used, perfect sets is somewhat topological and deals with the standard topology on Euclidean space. So I don't think it's fair to classify this as a Set Theoretic theorem (or at least not as solely a Set Theoretic theorem). $\endgroup$
    – MLevi
    Commented Nov 4, 2009 at 20:24
  • $\begingroup$ If I remember right, Cantor referred to cardinality as the "power" of a set at that time, in either German or French. (Hazy memories of reading some of his original letters in a history-of-logic compendium once.) $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented May 8, 2010 at 2:34
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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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    $\begingroup$ 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). $\endgroup$
    – lhf
    Commented Oct 30, 2009 at 9:18
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    $\begingroup$ 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. $\endgroup$
    – Steven Gubkin
    Commented Nov 12, 2009 at 20:04
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Euler's theorem V-E+F=2, can be regarded as a first theorem in graph theory, or in the theory of convex polytopes, and probably 0th theorem in algebraic topology and other fields.

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Algebraic topology: Fundamental group of S^1.

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    $\begingroup$ 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. $\endgroup$
    – Sam Derbyshire
    Commented Oct 25, 2009 at 2:47
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    $\begingroup$ Having just taught this result, I definitely agree with Qiaochu! Sam, if you can write down a "tautological" proof of this fact, I'd like to see it! $\endgroup$
    – HJRW
    Commented Nov 5, 2009 at 1:00
  • $\begingroup$ To be precise, I suppose I mean a proof that doesn't involve homotopy lifting. $\endgroup$
    – HJRW
    Commented Nov 5, 2009 at 1:01
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    $\begingroup$ 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). $\endgroup$
    – Sean Tilson
    Commented Mar 5, 2010 at 21:05
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    $\begingroup$ This is the most important computation in algebraic topology, in my opinion. Everything else ultimately derives from it. $\endgroup$
    – Jeff Strom
    Commented Jul 23, 2010 at 18:06
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Knot theory: sufficiency of the Reidemeister moves.

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Commutative Algebra: The Hilbert Basis Theorem: If R is noetherian, so is R[x].

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Ring theory: If R is a UFD, then so is R[x].

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    $\begingroup$ For ring theory, I prefer R noetherian implies R[x] is. The Hilbert Basis Theorem. $\endgroup$
    – Charles Siegel
    Commented Oct 26, 2009 at 20:56
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    $\begingroup$ Post this as a separate answer so that it can get voted independently. $\endgroup$
    – lhf
    Commented Oct 30, 2009 at 9:20
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Category theory: The Yoneda lemma.

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    $\begingroup$ 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. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 24, 2009 at 20:36
  • $\begingroup$ I think I'm only now getting where you're going. You want the first result after all the basic tools have been introduced? $\endgroup$
    – Ilya Nikokoshev
    Commented Oct 24, 2009 at 20:48
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    $\begingroup$ 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. $\endgroup$
    – Anton Geraschenko
    Commented Oct 24, 2009 at 20:56
  • $\begingroup$ I think Anton said what I wanted to get across: the Yoneda lemma itself isn't non-trivial, but the philosophy of it is. $\endgroup$
    – Harrison Brown
    Commented Oct 24, 2009 at 21:28
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    $\begingroup$ Maybe it's more correct to say that Yoneda is the last trivial theorem? $\endgroup$
    – Harrison Brown
    Commented Oct 25, 2009 at 5:52
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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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    $\begingroup$ The first nontrivial statement in class field theory? $\endgroup$
    – James Cranch
    Commented Mar 1, 2012 at 15:35
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Differential equations: Picard's theorem on existence and uniqueness.

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Graph theory: necessary and sufficient conditions for the existence of an Eulerian walk/cycle

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Linear algebra: rank-nullity theorem.

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    $\begingroup$ Although of course, this is just a categorification of the corresponding statement for finite sets... $\endgroup$
    – Kim Morrison
    Commented Oct 24, 2009 at 21:42
  • $\begingroup$ Do you mean the statement that a set is the union of the preimages of the image elements under a map? I think both statements are categorifications of the existence of addition, but linear categorifications are intrinsically more powerful. $\endgroup$
    – S. Carnahan ♦
    Commented Oct 24, 2009 at 23:04
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    $\begingroup$ 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 :) $\endgroup$
    – GMRA
    Commented Oct 25, 2009 at 2:08
  • $\begingroup$ I've been teaching some linear algebra this term. It's really satisfying seeing pupils realise just what a powerful tool this theorem is. (Powerful, that is, in the context of first year lin alg...) $\endgroup$
    – Tom Smith
    Commented May 23, 2010 at 20:52
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    $\begingroup$ 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/…) $\endgroup$
    – Victor Protsak
    Commented Jul 24, 2010 at 5:33
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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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    $\begingroup$ 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. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 24, 2009 at 20:39
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    $\begingroup$ 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. $\endgroup$
    – Michael Lugo
    Commented Oct 24, 2009 at 21:59
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    $\begingroup$ 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. $\endgroup$
    – Chad Groft
    Commented Feb 22, 2010 at 17:31
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Probability theory: the Central Limit Theorem?

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    $\begingroup$ I would consider the law of large numbers non-trivial, and that was covered in my probability courses long before the CLT. $\endgroup$
    – Kevin P. Costello
    Commented Oct 25, 2009 at 0:19
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Social choice theory: Arrow's Impossibility Theorem.

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  • $\begingroup$ This is the first time I’ve heard of Social Choice Theory, which is why I have to upvote this answer. $\endgroup$
    – k.stm
    Commented Feb 19, 2014 at 9:12
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Functional Analysis: the Hahn-Banach Theorem.

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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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Functional analysis: the open mapping theorem.

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  • $\begingroup$ there's a case for saying "Banach-Steinhaus theorem" or "uniform boundedness", which are somehow the deepest part of (the proof of) the open mapping theorem. But yes, this one gets a vote from me. $\endgroup$
    – Yemon Choi
    Commented Oct 25, 2009 at 20:56
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Computer science: sort requires n * log n

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    $\begingroup$ Nitpick: Comparison based sort is \Omega(n*log n). $\endgroup$
    – Steven Sam
    Commented Oct 24, 2009 at 20:16
  • $\begingroup$ Comparison as in bubble sort (nn) or as in some other sort? I mean *any sorting algorithm requires asymptotically at least C * n * log n. $\endgroup$
    – Ilya Nikokoshev
    Commented Oct 24, 2009 at 20:21
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    $\begingroup$ That is only true for comparison-based sorting algorithms. There are non comparison-based sorting algorithms if your data consists of numbers or alphabets, or things like that. For example, radix sort: en.wikipedia.org/wiki/Radix_sort $\endgroup$
    – Steven Sam
    Commented Oct 24, 2009 at 20:35
  • $\begingroup$ Well, it's simply about the way you define your problem; other sorts have input defined in a different way. "Hence, radix sort does not really beat O(n log n) time, it only appears to do so because the range of keys is implicitly limited by the size of k." (from wikipedia) $\endgroup$
    – Ilya Nikokoshev
    Commented Oct 24, 2009 at 20:46
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    $\begingroup$ 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. $\endgroup$
    – Konrad Swanepoel
    Commented Nov 11, 2009 at 10:09
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Finite group representation theory: the fact that the irreducible characters form a basis for the class functions.

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Symplectic Geometry: Darboux's theorem

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  • $\begingroup$ Does this predate Liouville's theorem that Hamiltonian flows are symplectomorphisms (according to wikipedia he's only attributed for the volume preserving part)? Liouville was about 30 years older. $\endgroup$
    – PVAL
    Commented May 2, 2016 at 9:37
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Noncommutative ring theory: The Artin-Wedderburn Theorem.

A ring R is semisimple if and only if it is isomorphic to a finite direct product of matrix rings over division rings.

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  • $\begingroup$ Being R a unital ring. $\endgroup$
    – Jose Brox
    Commented Nov 27, 2009 at 2:35
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Although the Gauss-Bonnet theorem was cited for differential geometry of surfaces, I really think that the first striking result in this subject is Gauss's Theorema Egregium, which is not obvious from the definition of Gaussian curvature (which makes explicit reference to the ambient space). But the Gauss-Bonnet theorem is certainly the first really deep theorem one encounters in differential geometry.

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    $\begingroup$ Theorema Egregium is certainly very deep! Without Theorema Egregium to back it up, the Gauss-Bonnet theorem isn't very meaningful, and of course, it comes later. $\endgroup$
    – Victor Protsak
    Commented May 23, 2010 at 21:24
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operator algebras: the Gelfand transform is an isomorphism for commutative $C^*$-algebras.

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Riemann Surfaces: Riemann-Roch Theorem

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

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Algebraic topology: Poincaré Duality

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Euclidean geometry: a triangle on a semicircle has a right angle.

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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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    $\begingroup$ Now, isn't Cauchy-Davenport already quite interesting? $\endgroup$
    – darij grinberg
    Commented May 23, 2010 at 22:05
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