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

Differential Geometry (of surfaces, say): the Gauss-Bonnet theorem.

Answer 2

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

Answer 3

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.

Answer 4

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.

Answer 5

Points on an elliptic curve form an abelian group, accredited to Fermat.

This should appear as an example when one introduces group theory. One can easily state many non-trivial facts, like the rational points form a finitely generated subgroup, whose torsion is known (Mazur), and whose rank is the subject of Birch-Swinnerton-Dyer conjecture. (That would also be a nice example of the classification theorem of finitely generated abelian groups)

Answer 6

In geometric probability: Buffon's noodle (and needle).

Answer 7

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.

Answer 8

Harmonic Analysis: Plancherel's theorem

Answer 9

The first interesting theorem in Differential Algebra is...

Liouville´s condition for integration of elementary functions in finite terms.

Answer 10

Algebraic number theory: Hilbert 90.

Answer 11

Complex analysis: Riemann mapping theorem.

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

Answer 12

The representation theory of compact groups: The Peter-Weyl Theorem.

Answer 13

Poset theory: Dilworth's theorem.

Answer 14

Rings with polynomial identities (PI rings): The Amitsur Levitzki's theorem http://gilkalai.wordpress.com/2009/05/12/the-amitsur-levitski-theorem-for-a-non-mathematician/

Answer 15

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

Answer 16

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)

Answer 17

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

Answer 18

Commutative algebra: primary decomposition.

Answer 19

Analysis/Topology. The closed interval [a,b] is compact.

Answer 20

Algebra: Classification of finite abelian groups

Answer 21

Nonlinear programming/Optimization: The Karush-Kuhn-Tucker conditions

Answer 22

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

Answer 23

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.

Answer 24

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.

Answer 25

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.

Answer 26

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?

Answer 27

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.

Answer 28

Real Analysis: The function $t \mapsto \exp(it)$, defined by a certain power series, is periodic with a period of about 6.4.

Answer 29

Algebraic geometry: points are prime ideals.

Answer 30

Fractional or arbitrary order calculus - The derivation of the known equation by Louisville to show that classical calculus is a special case of fractional calculus, in a paper written in 1832.

This was not a rigorous proof ala Euclid, but it was important to prove a concept that was known since at least 1695.