What is the first interesting theorem in (insert subject here)? 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.
 A: Combinatorics: the nth Catalan number is (2n choose n)/(n+1)
A: Algebraic topology: Poincaré Duality
A: Euclidean geometry: a triangle on a semicircle has a right angle.
A: 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.
A: 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.
A: In mathematical logic, Gödel's incompleteness theorem.
A: Homotopy Theory: the Hopf Fibration?
A: Differential Geometry (of surfaces, say): the Gauss-Bonnet theorem.
A: In analytic number theory: Euler's proof of the infinitude of primes, using the divergence of the harmonic series.
A: Algebraic number theory: Hilbert 90.
A: 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)
A: Poset theory:  Dilworth's theorem.
A: In geometric probability: Buffon's noodle (and needle).
A: 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/
A: 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.
A: The first interesting theorem in Differential Algebra is...
Liouville´s condition for integration of elementary functions in finite terms.
A: Harmonic Analysis: Plancherel's theorem
A: 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.
A: Complex analysis: Riemann mapping theorem.
(Easier candidates include: Liouville's theorem, Cauchy's integral formula, Picard theorems.)
A: The representation theory of compact groups: The Peter-Weyl Theorem.
A: Finite group theory:  I would say Lagrange's theorem, that the order of a subgroup divides the order of the group.  Certainly it's prior to the Sylow theorems, certainly it has content.
A: 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)
A: Commutative algebra:  primary decomposition.
A: Analysis/Topology. The closed interval [a,b] is compact.
A: 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?
A: Algebra: Classification of finite abelian groups
A: The uncountability of the reals.  This is such a classic fact that I'm not sure how to classify it.  Set theory, perhaps.  
A: Algebraic Geometry: Bezout's Theorem.  It's also good for selling what algebraic geometry is to people who've never heard of it before.
A: Euler's theorem V-E+F=2, can be regarded as a first theorem in graph theory, or in the theory of convex polytopes, and probably 0th theorem in algebraic topology and other fields. 
A: Algebraic topology: Fundamental group of S^1.
A: Knot theory: sufficiency of the Reidemeister moves.
A: Commutative Algebra: The Hilbert Basis Theorem: If R is noetherian, so is R[x].
A: Ring theory: If R is a UFD, then so is R[x].
A: Category theory: The Yoneda lemma.
A: Quadratic reciprocity feels to me like a "first nontrivial statement" without an obvious branch of mathematics. Not number theory -- there are things like the infinitude of the primes, and "algebraic number theory" doesn't seem quite right either...
A: Differential equations: Picard's theorem on existence and uniqueness.
A: Graph theory: necessary and sufficient conditions for the existence of an Eulerian walk/cycle
A: Linear algebra: rank-nullity theorem.
A: Combinatorics: counting the number of derangements of [n].
A: Banach algebras: Gelfand's proof of the Wiener lemma for l1(Z) 
A: Nonlinear programming/Optimization: The Karush-Kuhn-Tucker conditions
A: In real analysis, I would say The intermediate value theorem.
A: 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.
A: 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.
A: 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.
A: Real Analysis: The function $t \mapsto \exp(it)$, defined by a certain power series, is periodic with a period of about 6.4.
A: Number theory:  Different undergraduate textbooks approach the subject differently, of course. But the irrationality of the square root of 2 and the infinitude of primes are contentful theorems that are certainly very early historically, and also very early in at least some textbook treatments of the subject.
A: Probability theory: the Central Limit Theorem?
A: Social choice theory: Arrow's Impossibility Theorem.
A: Functional Analysis: the Hahn-Banach Theorem.
A: Theory of Computation: The halting problem
Set Theory: Cardinality of a set is strictly less than its powerset
Not sure which field (one might learn it in an intro real analysis course): Reals are uncountable
Complexity theory: The Time and Space Hierarchy theorems
One might learn these results in different courses, but they're all the same beautiful idea: Cantor's diagonalization.
A: Functional analysis: the open mapping theorem.
A: Computer science: sort requires n * log n 
A: Symplectic Geometry: Darboux's theorem
A: Finite group representation theory: the fact that the irreducible characters form a basis for the class functions.
A: Noncommutative ring theory: The Artin-Wedderburn Theorem.
A ring R is semisimple if and only if it is isomorphic to a finite direct product of matrix rings over division rings.
A: Although the Gauss-Bonnet theorem was cited for differential geometry of surfaces, I really think that the first striking result in this subject is Gauss's Theorema Egregium, which is not obvious from the definition of Gaussian curvature (which makes explicit reference to the ambient space). But the Gauss-Bonnet theorem is certainly the first really deep theorem one encounters in differential geometry.
A: operator algebras: the Gelfand transform is an isomorphism for commutative $C^*$-algebras.
A: Riemann Surfaces: Riemann-Roch Theorem
A: Algebraic geometry: points are prime ideals.
A: 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.
A: 3-manifolds, existence and uniqueness of decomposition into irreducible pieces in the connected sum (Kneser, Milnor-Wallace)
A: Differential Geometry: Rank Theorem
A: Ergodic theory: Poincare's recurrence theorem
http://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem
A: Complex analysis: Hadamard's factorization theorem. 
A: Symbolic dynamics: There is a unique minimal right resolving presentation for an irreducible sofic shift.
A: 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.
A: 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.
A: 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.
A: ML：Gödel incompleteness theorem
A: Plane geometry, either Euclid Bk 1, Prop 47 (Pythagoras' theorem), or the nine-point circle theorem.  I can't decide.
A: 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)
A: Linear Algebra: The Principal Axis Theorem. Quadratic forms over reals have a signature, i.e. after a change of coordinates, are of the form $(x_1^2+\cdots+x_n^2)-(y_1^2+\cdots + y_m^2)$.
A: Fourier Theory:The Fast Fourier Transform
A: Number Theory -- Mordell-Weil Theorem
A: Number Theory: Kronecker-Weber (Every abelian extension of Q is contained in a cyclotomic extension.)
A:   fundamental theorem of algebra
  fundamental theorem of calculus

