Foundations

Over a field F, f=gq+r has unique q,r for nonzero g, with r=0 or deg r<deg g.

A nonconstant polynomial over F is irreducible if it cannot factor into two positive-degree polynomials over F.

The sum of squares of k independent standard normal variables has chi-square law χ²ₖ=Gamma(k/2,1/2), with mean k and variance 2k.

If Z∼N(0,1) and U∼χ²ν are independent, Z/√(U/ν) has tν law.

Recompute statistic T on many size-n simulated data sets.

A fixed finite-tape deterministic Turing machine running in time t(n)≥n can be simulated by a single-tape machine in O(t(n)²) time by storing tape contents and head markers and scanning them for each simulated step.

A deterministic decider's time cost counts steps and its space cost counts used storage cells; worst-case bounds maximize over inputs of the same encoded length.

For eventually nonnegative terms aₙ≤bₙ, convergence of Σbₙ implies convergence of Σaₙ; divergence of Σaₙ implies divergence of Σbₙ.

If |aₙ₊₁/aₙ|→ρ, or |aₙ|¹⁄ⁿ→ρ, then Σaₙ converges absolutely for ρ<1 and diverges for ρ>1.

If differentiable real fₙ on (a,b) converge pointwise to f and fₙ′ converge uniformly to g, then f is differentiable and f′=g.

If Riemann-integrable fₙ on a finite interval [a,b] converge uniformly to f, then f is Riemann integrable and ∫ₐᵇfₙ dx→∫ₐᵇf dx.

For Σaₙ(x−c)ⁿ the radius R is 1/lim sup|aₙ|¹⁄ⁿ, with 1/0=∞ and 1/∞=0.

Disjoint finite alternatives have total count equal to the sum of their counts.

For finite A,B, |A∪B|=|A|+|B|−|A∩B|.

From n distinct objects choose k without repetition.

A directed network flow respects nonnegative capacities and conserves flow at vertices other than source s and sink t.

Starting with zero flow, find a shortest residual s–t path by breadth-first search and augment by its minimum residual capacity, updating forward and reverse capacities.

For N complex coefficients aⱼ define yₖ=Σ_(j=0)^(N−1)aⱼωᴶᴷ, with ω=e²πⁱ⁄ᴺ.

For N a power of two, split polynomial coefficients into even and odd indices, recursively transform each half, and combine values using p(z)=p_even(z²)+zp_odd(z²).

A unit has a multiplicative inverse.

For integral domain R, fractions a/b with b≠0 are equivalence classes under a/b=c/d iff ad=bc.

For a commutative ring with identity and proper ideal I, I is prime iff ab∈I implies a∈I or b∈I, equivalently R/I is an integral domain.

A field has characteristic zero if no positive n satisfies n·1=0; otherwise its characteristic is a prime p.

In an integral domain irreducibles are nonzero nonunits that cannot split into two nonunits; primes divide a product only by dividing a factor.

A Euclidean domain has a nonnegative integer size d on nonzero elements permitting b=aq+r with r=0 or d(r)<d(a), and d(a)≤d(ab).

For fields F⊆K, degree [K:F] is dim_F K.

An algebraic α satisfies a nonzero polynomial over F.

A finite field has pⁿ elements for a prime p and n≥1; for every such size one exists uniquely up to isomorphism.

A splitting field is generated by all roots of a polynomial; a separable polynomial has distinct roots.

Along a piecewise C¹ regular path r(t), scalar integration is ∫C f ds=∫f(r(t))‖r′(t)‖dt and vector integration is ∫C F·dr=∫F(r(t))·r′(t)dt.