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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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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

78 Answers 78

Number Theory -- Mordell-Weil Theorem

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That's the first interesting/non-trivial result in Number Theory? –  Todd Trimble Mar 1 '12 at 15:59

Fourier Theory:The Fast Fourier Transform

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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

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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Algebra: Classification of finite abelian groups

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Differential Geometry: Rank Theorem

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Order / Lattice Theory (?): The Boolean Prime Ideal Theorem (BPI). It is strictly weaker than the axiom of choice but equivalent to Tychonovs theorem (EDIT: of course the version where the spaces are Hausdorff, compact) therefore underlying all of functional analysis.

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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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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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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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Euler's theorem

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Which one? There are many theorems by Euler... –  lhf May 23 '10 at 22:11
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This reminds me the talk given by Jean-Pierre Serre ''How to write badly mathematics''. He gave an example for a title: ''On a theorem of Euler''. He suggested also to include a reference to ''L. Euler. Opera omnia''. –  Denis Serre Apr 26 '11 at 17:15
  fundamental theorem of algebra
  fundamental theorem of calculus
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In what subject is the fundamental theorem of algebra the first interesting theorem? –  Jonas Meyer Jan 15 '10 at 9:30
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Complex analysis? –  darij grinberg May 23 '10 at 22:04

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

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Linear Algebra: The Principal Axis Theorem. Quadratic forms over reals have a signature, i.e. after a change of coordinates, are of the form $(x_1^2+\cdots+x_n^2)-(y_1^2+\cdots + y_m^2)$.

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The first interesting theorem in Non Euclidian Geometry is...

This isn´t a proof, but it´s the object of the study fisically represented: paper model of hyperbolic plane (by Bill Thurston). I´m lovin it! ...and of course, the models of Klein and of Poincaré of hyperbolic plane (My favorite is the half plane)

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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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Harmonic Analysis: Plancherel's theorem

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Plane geometry, either Euclid Bk 1, Prop 47 (Pythagoras' theorem), or the nine-point circle theorem. I can't decide.

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Number Theory: Kronecker-Weber (Every abelian extension of Q is contained in a cyclotomic extension.)

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Mine is Fermat's criterion for a number to be expressed as the sum of two squares. –  lhf Dec 15 '09 at 12:32

3-manifolds, existence and uniqueness of decomposition into irreducible pieces in the connected sum (Kneser, Milnor-Wallace)

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ML:Gödel incompleteness theorem

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

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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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. –  Victor Protsak May 23 '10 at 21:24

In Euclid's Elements the first proposition is the construction of the equilateral triangle. An interesting result, perhaps, but also important as a beginning of an explanation of what "proof" shall mean.

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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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In algebraic number theory, at least in terms of a serious interesting result with a difficult proof, I'd say the Tchebotarev Density Theorem.

More elementary would be Sum(ef)=n and/or properties of decomposition and inertia groups. Also Dirichlet's Unit Theorem, depending on what order you work in.

In analytic number theory, I would say the Prime Number Theorem, at least if we're looking for the first difficult, interesting result.

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

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