(mathematics) A theorem showing that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups; one of the foundational results of anabelian geometry.

(mathematics) A theorem stating that for each positive integer d there is a finite set Y_d of prime numbers, such that if p is any prime not in Y_d then every homogeneous polynomial of degree d over the p-adic numbers in at least d²+1 variables has a nontrivial zero.

(mathematics) A theorem stating that a sequence of functions that is locally of bounded total variation and uniformly bounded at a point has a convergent subsequence.

A theorem dealing with deterministic ordinal electoral systems that choose a single winner, stating that for every voting rule, one of the following three things must hold: (i) the rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or (ii) the rule limits the possible outcomes to two alternatives only; or (iii) the rule is susceptible to tactical voting: in certain conditions some voter's sincere ballot may not defend their opinion best.

(mathematics) A result about injectivity and surjectivity of polynomials, often given as this special case: If P is an injective polynomial function from an n-dimensional complex vector space to itself then P is bijective. The full theorem generalizes to any algebraic variety over an algebraically closed field.

(mathematics) A theorem about packing congruent rectangular bricks into larger rectangular boxes so that no space is left over. It states that a "harmonic brick" (one in which each side length is a multiple of the next smaller side length) can only be packed into a box whose dimensions are multiples of the brick's dimensions.

(geometry) A theorem stating that, given a finite number of points in the Euclidean plane, either all the points lie on a single line, or there is at least one line which contains exactly two of the points.

In dynamical systems theory, a theorem stating that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also known n first integrals of motion that are independent and in involution, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time. Thus the equations of motion for the system can be solved in quadratures if the canonical transform is explicitly known.

(mathematics) A particular refinement of the prime number theorem and of Dirichlet's theorem on primes in arithmetic progressions.

(number theory) the theorem that states that every integer greater than one is uniquely expressible as a product of prime numbers, which is called its prime factorization