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If the prime $p$ does not divide $n$, then $X^n-1$ is square-free over ${\mathbb{F}}_p$ because its derivative $n{X}^{n-1}$ is coprime to it. Its factor ${\Phi }_n$ is therefore separable modulo $p$.

Partitioning the roots of $X^n-1$ by their exact orders givesInductively, the product over proper divisors is a monic polynomial in $\mathbb{Z}[X]$. Division of $X^n-1$ by this monic polynomial stays in $\mathbb{Z}[X]$ and has zero remainder, proving that ${\Phi }_n\in \mathbb{Z}[X]$.

Given a primitive $n$th root $\alpha$, a BCH code of design distance $\delta$ has a generator polynomial vanishing at $\delta -1$ consecutive powers ${\alpha }^b,\dots ,{\alpha }^{b+\delta -2}$.

Over ${\mathbb{F}}_2$,Its eight monic divisors generate the eight cyclic codes of length seven: the whole-space, even-weight, two Hamming, two simplex, repetition, and zero codes.

The dual of a cyclic code is cyclic. If $g(X)h(X)=X^n-1$, then the dual generator is the monic reciprocal polynomial $h^{\ast }(X)$.

The unique monic divisor $g(X)$ of $X^n-1$ generating a cyclic code is its generator polynomial. The code has dimension $n-\mathrm{deg}g$.

A maximum cut has greatest size among all cuts of a finite graph.

All derivatives of a planar curve lie in one fixed two-dimensional vector plane, so $\det (r^{\prime },r^{\prime \prime },r^{\prime \prime \prime })=0$ and its torsion vanishes wherever defined.

The total curvature of a regular curve isIt equals the length traced by the unit tangent on the unit sphere, so lifting a planar circle into a helix can reduce total curvature even when both curves make one revolution around the same axis.

For a unit-speed curve $\gamma$, the unit tangent is $T={\gamma }^{\prime }$ and the curvature is $\kappa =|T^{\prime }|$. It measures the rate at which the tangent direction turns.

The class of hereditarily finite sets is the setEquivalently, these are the sets with finite transitive closure, or the sets contained in a finite transitive set.

The transitive closure $\mathrm{TC}(x)$ is the least transitive set containing every member of $x$. It is obtained by taking the union of all finite iterates of the union operation starting from $x$. This set-theoretic construction uses the membership binary relation; transitive closure of a relation instead closes an arbitrary binary relation under composition.

For $g(X)=X^3-5X^2+11X-12$, the reductions areandBoth quadratic factors are irreducible, soand

The elementsform an integral basis. Relative to this basis, multiplication by $\beta$ has integer matrixThe basis discriminant is one quarter of the discriminant $-4\cdot 307$ of $(1,\alpha ,{\alpha }^2)$, hence is the square-free integer $-307$.

If a simple graph has $n$ vertices and $e\ge 4n$ edges, every plane drawing has at leastcrossings. Apply the linear planar bound to the random induced subgraph obtained by retaining vertices independently with probability $4n/e$.

The number operator $N=A^{†}A$ satisfiesIt is self-adjoint and positive semidefinite.

A stratified Cox model gives each stratum its own baseline hazard while sharing regression coefficients.

The Cox partial likelihood conditions each event on its risk set, eliminating the baseline hazard.

An action of $G$ on $X$ is a covering space action when every point has a neighbourhood $U$ disjoint from $gU$ for every nonidentity $g$. The quotient map is then locally a covering map, although without stronger properness assumptions its quotient need not be Hausdorff.

For a covering $p:(\tilde{X},{\tilde{x}}_0)\to (X,x_0)$ and a based map $f:(Y,y_0)\to (X,x_0)$ from a path-connected locally path-connected space, a based lift $\tilde{f}$ exists exactly when