Question

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.

Answer 1

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

Answer 2

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

Answer 3

Differential Geometry: Rank Theorem

Answer 4

Complex analysis: Hadamard's factorization theorem.

Answer 5

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

Answer 6

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.

Answer 7

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.

Answer 8

ML：Gödel incompleteness theorem

Answer 9

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

Answer 10

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)

Answer 11

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

Answer 12

  fundamental theorem of algebra
  fundamental theorem of calculus

Answer 13

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.

Answer 14

Fourier Theory:The Fast Fourier Transform

Answer 15

Ergodic theory: Poincare's recurrence theorem http://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem

Answer 16

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

Answer 17

Number Theory -- Mordell-Weil Theorem

Answer 18

Euler's theorem