Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question

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

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

3  
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
show 4 more comments

78 Answers 78

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

share|improve this answer
show 1 more comment

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

share|improve this answer
add comment

Differential Geometry: Rank Theorem

share|improve this answer
add comment

Complex analysis: Hadamard's factorization theorem.

share|improve this answer
add comment

Symbolic dynamics: There is a unique minimal right resolving presentation for an irreducible sofic shift.

share|improve this answer
add comment

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.

share|improve this answer
show 1 more comment

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.

share|improve this answer
add comment

ML:Gödel incompleteness theorem

share|improve this answer
add comment

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

share|improve this answer
add comment

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)

share|improve this answer
add comment

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

share|improve this answer
show 3 more comments

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.

share|improve this answer
show 2 more comments

Fourier Theory:The Fast Fourier Transform

share|improve this answer
add comment

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

share|improve this answer
2  
Mine is Fermat's criterion for a number to be expressed as the sum of two squares. –  lhf Dec 15 '09 at 12:32
show 2 more comments
  fundamental theorem of algebra
  fundamental theorem of calculus
share|improve this answer
2  
In what subject is the fundamental theorem of algebra the first interesting theorem? –  Jonas Meyer Jan 15 '10 at 9:30
1  
Complex analysis? –  darij grinberg May 23 '10 at 22:04
add comment

Number Theory -- Mordell-Weil Theorem

share|improve this answer
3  
That's the first interesting/non-trivial result in Number Theory? –  Todd Trimble Mar 1 '12 at 15:59
add comment

Euler's theorem

share|improve this answer
4  
Which one? There are many theorems by Euler... –  lhf May 23 '10 at 22:11
7  
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
add comment

Not the answer you're looking for? Browse other questions tagged or ask your own question.