structure
55 theorems carry this thread across the degree.
The principle of mathematical induction
If a statement holds for 1 and its truth at n forces its truth at n+1, it holds for every natural number.
The well-ordering principle
Every non-empty set of natural numbers has a least element, and this is equivalent to induction.
The pigeonhole principle
If n+1 objects are placed in n boxes, some box holds at least two.
The irrationality of √2
No ratio of integers squares to 2; the proof is the classic argument by contradiction.
The infinitude of the primes
There is no largest prime; assuming a finite list yields a contradiction.
The uncountability of the reals
No list can enumerate all real numbers — Cantor's diagonal argument.
Cantor's theorem
A set never has the same cardinality as its power set: |P(A)| > |A|.
The rank–nullity theorem
For a linear map, rank plus nullity equals the dimension of the domain.
The invertible matrix theorem
A dozen conditions on a square matrix are all equivalent to invertibility.
Multiplicativity of the determinant
det(AB) = det(A)det(B).
The Cauchy–Schwarz inequality
The inner product of two vectors is bounded by the product of their norms.
The Gram–Schmidt process
Any basis can be turned into an orthonormal one.
Cramer's rule
Solutions of a square linear system as ratios of determinants.
The binomial theorem
An expansion of (x+y)^n in terms of binomial coefficients.
The inclusion–exclusion principle
The size of a union from the sizes of intersections.
The handshaking lemma
In any graph the sum of degrees is twice the number of edges.
Fermat's little theorem
a^p ≡ a (mod p) for prime p.
The Chinese remainder theorem
Congruences with coprime moduli have a unique joint solution.
Euler's theorem on circuits
A connected graph has an Eulerian circuit iff every vertex has even degree.
Invariance of dimension
Every basis of a vector space has the same cardinality.
The spectral theorem
A self-adjoint operator has an orthonormal basis of eigenvectors.
The Cayley–Hamilton theorem
Every matrix satisfies its own characteristic polynomial.
The Jordan normal form
Every operator over C is similar to a direct sum of Jordan blocks.
The singular value decomposition
Any matrix factors as a rotation, a scaling, and a rotation.
The orthogonal projection theorem
Best approximation in an inner product space is orthogonal projection.
Lagrange's theorem
The order of a subgroup divides the order of the group.
The orbit–stabiliser theorem
Orbit size times stabiliser size equals the group order.
The first isomorphism theorem
The image of a homomorphism is the quotient by its kernel.
Cauchy's theorem
If a prime divides the group order, an element of that order exists.
The Sylow theorems
Existence, conjugacy, and counting of maximal p-subgroups.
Structure of finite abelian groups
Every finite abelian group is a product of cyclic groups.
Every PID is a UFD
Principal ideal domains have unique factorisation.
Eisenstein's criterion
A prime-based test for irreducibility of polynomials.
The tower law
Degrees of field extensions multiply.
The fundamental theorem of Galois theory
Subfields correspond to subgroups of the Galois group.
Classification of finite fields
There is exactly one field of each prime-power order.
Insolvability of the quintic
No general radical formula solves degree-five equations.
The Hahn–Banach theorem
Bounded functionals extend without increasing norm.
The open mapping theorem
A surjective bounded operator between Banach spaces is open.
The closed graph theorem
A closed-graph operator between Banach spaces is bounded.
The uniform boundedness principle
Pointwise-bounded families of operators are uniformly bounded.
The Riesz representation theorem
Every bounded functional on a Hilbert space is an inner product.
Ramsey's theorem
Complete disorder is impossible in large enough structures.
Hall's marriage theorem
A matching exists iff every set of vertices has enough neighbours.
The max-flow min-cut theorem
Maximum flow equals minimum cut capacity.
Kuratowski's theorem
A graph is planar unless it contains K5 or K3,3.
Turán's theorem
The maximum edges in a graph with no large clique.
The fundamental group
Loops up to homotopy form a group invariant of the space.
The Seifert–van Kampen theorem
The fundamental group of a union from those of its pieces.
The Brouwer fixed-point theorem
Every continuous self-map of a disc has a fixed point.
Invariance of the Euler characteristic
V−E+F is a topological invariant.
Maschke's theorem
Representations of finite groups over C are completely reducible.
Schur's lemma
Morphisms between irreducibles are zero or isomorphisms.
Orthogonality of characters
Irreducible characters form an orthonormal set.
Burnside's counting lemma
Orbits counted by average fixed points.