## What is the first interesting theorem in (insert subject here)? [closed]

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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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 2012 at 16:03
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## closed as no longer relevant by Felipe Voloch, S. Sra, Bill Johnson, Todd Trimble, Qiaochu YuanMar 1 2012 at 16:41

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

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

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.

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Excluding, of course, the result of Lam, Thiel and Swiercz on the plane of order 10. In combinatorial design theory more generally, there are certainly other non-trivial results. – Will Orrick Oct 25 2009 at 0:00
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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)

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I'd call this an example in group theory but a theorem in algebraic geometry. – Qiaochu Yuan Oct 27 2009 at 2:38
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In geometric probability: Buffon's noodle (and needle).

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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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Harmonic Analysis: Plancherel's theorem

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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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Algebraic number theory: Hilbert 90.

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Complex analysis: Riemann mapping theorem.

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

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At least for me, taking this course as an undergraduate, it was the Cauchy integral formula for sure. – JSE Oct 24 2009 at 22:36
he Cauchy integral formula may be under-appreciated these days because we've moved it up early in the curriculum. We think it's elementary because it's presented early on. But historically it came after much of the material now in a complex analysis course. It was moved up precisely because it is so powerful and makes the proofs of other theorems easier. – John D. Cook Oct 26 2009 at 4:39
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The representation theory of compact groups: The Peter-Weyl Theorem.

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Poset theory: Dilworth's theorem.

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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/

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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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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)

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I agree it's non-trivial if you try to prove it purely algebraically, but the proof using the unitary trick on the associated simply connected compact group is pretty easy to digest. – Dinakar Muthiah Oct 25 2009 at 5:09
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Combinatorics: counting the number of derangements of [n].

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I think I would disagree, since counting derangements is just a standard application of inclusion-exclusion, which is pretty trivial. – Harrison Brown Oct 24 2009 at 21:30
Combinatorics doesn't really fall under the purvey of this question, since it's both relatively non-axiomatic and highly non-linear. And for what it's worth, I consider inclusion-exclusion highly nontrivial, at least conceptually (as Mobius inversion). – Qiaochu Yuan Oct 24 2009 at 21:32
I think this answer is reasonable. But this implicitly requires that permutations be one of the first objects you look at. As Qiaochu has pointed out, we don't necessarily have to make this choice. – Michael Lugo Oct 24 2009 at 21:56
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Commutative algebra: primary decomposition.

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I'd argue for Nakayama on the same grounds as the Yoneda lemma, but again that's of a rather different nature. – Harrison Brown Nov 27 2009 at 4:07
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Analysis/Topology. The closed interval [a,b] is compact.

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Algebra: Classification of finite abelian groups

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

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

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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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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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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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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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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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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 2012 at 14:43

Algebraic geometry: points are prime ideals.

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I would call this more a modification of the definition of "point" than anything else. But I would agree that something like "the Nullstellensatz" is a great answer here. – Qiaochu Yuan Oct 24 2009 at 20:09
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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.

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