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

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I wish the system was smart enough to understand that example = examples and soft-question = soft-questions (and topos = topoi :) ). – Ilya Nikokoshev Oct 24 '09 at 20:34
@Qiaochu: If there is no question with a given tag, the tag will disappear on its own within about a day. If you find two tags that should really be the same, flag the post for moderator attention with a comment explaining that the tags should be merged. @ilya: the site doesn't understand the meanings of the tags, so it would probably cause trouble if it tried to automatically merge tags. But you usually don't write the complete name of the tag anyway; you just type the first few letters and then you're shown a list of tags that match. – Anton Geraschenko Oct 24 '09 at 20:50
@Anton, the object = objects idea seem to be easy to implement to me -- is there an example where this merging makes no sense? – Ilya Nikokoshev Oct 24 '09 at 23:20
Is anyone other than me in favor of changing the word "non-trivial" to "interesting" in the title? I, for one, would feel much less intimidated. – S. Carnahan Oct 26 '09 at 9:45
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

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

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(I suppose someone will point out that this rests on the Gelfand-Mazur theorem, but I maintain that Wiener's lemma is more arresting and grabs the attention better - hence is more "interesting" in that sense. Obviously the Gelfand-Mazur theorem lies deeper and is in that sense more "interesting") – Yemon Choi Nov 11 '09 at 10:16

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

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Differential Geometry: Rank Theorem

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

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

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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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I might add the finiteness of the class number. Or if you're looking for theorems which show how the machinery can be used, I would add some of the elementary Diophantine problems which can be solved using finiteness of the class number. – David Corwin May 9 '10 at 15:24

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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ML：Gödel incompleteness 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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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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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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This is called Sylvester's Rule. – Harry Gindi Dec 13 '09 at 14:46
Sylvester's Law, rather. – Harry Gindi Dec 13 '09 at 14:47
I'd say it shares its place with Cayley-Hamilton. – darij grinberg May 23 '10 at 22:06

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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Tychonoff's theorem is equivalent to choice. Perhaps you mean Tychonoff for compact Hausdorff spaces? – Andrés E. Caicedo Jul 7 '10 at 16:06
That is what I meant. – santker heboln Jul 21 '10 at 10:56

Fourier Theory:The Fast Fourier Transform

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

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This is a wonderful theorem but it's hardly the first interesting theorem in Number Theory. – lhf Dec 1 '09 at 12:40
It is really up to personal taste :) I understand there are other more elementary ones like finiteness theorems for class number, etc. But I really find Kronecker-Weber the first interesting thing to me. Your favorite interesting theorem? – user2148 Dec 8 '09 at 16:53
Mine is Fermat's criterion for a number to be expressed as the sum of two squares. – lhf Dec 15 '09 at 12:32
  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
Complex analysis? – darij grinberg May 23 '10 at 22:04

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