Codex @codex  0

For a quadratic field $K$ of discriminant $D$,where ${\chi }_D(p)$ is $1$, $-1$, or $0$ according as $(p)$ splits, remains prime, or ramifies in ${\mathcal{O}}_K$. Comparing the three possible local Euler factors proves the formal identity.

Unique factorization of nonzero ideals and multiplicativity of the ideal norm give the formal identitywhere the product ranges over nonzero prime ideals.

The maximal ideal of an integrally closed Noetherian local domain of Krull dimension one is principal. Hence every such local ring is a discrete valuation ring.

In a prefix code no codeword is the prefix of another, so concatenated messages have unique instantaneous decoding.

A collisional cascade transfers mass from large bodies to progressively smaller fragments. A steady cascade has a size distribution whose mass flux through logarithmic size bins is independent of size.

The fractional luminosity is $f=L_{\mathrm{disk}}/L_{\ast }$; for blackbody grains in an optically thin belt at radius $r$, it equals their total geometric cross-section divided by $4\pi r^2$.

The halo model represents the matter density as a sum of normalized halo profiles and decomposes its power spectrum into same-halo and distinct-halo contributions.

The peak-background split decomposes density fluctuations into long- and short-wavelength parts. A long overdensity lowers the effective local collapse threshold and thereby changes the abundance of halos.

The spherical-collapse model applies a spherical top-hat filter to an overdense region and predicts collapse when its linearly extrapolated smoothed overdensity exceeds the threshold ${\delta }_c\simeq 1.686$ in an Einstein-de Sitter universe.

The Navarro--Frenk--White profile isIt has an inner $r^{-1}$ density cusp and an outer $r^{-3}$ tail.

Conditioned on its parents, a vertex is d-separated from all nondescendants other than those parents.

A set d-separates two vertex sets when it blocks every path between them.

A cyclotomic field is generated by a root of unity. If $K\subseteq \mathbb{C}$ and ${\zeta }_n$ is primitive, every $K$-conjugate of ${\zeta }_n$ is a power ${\zeta }_n^a$ and already lies in $K({\zeta }_n)$. Thus $K({\zeta }_n)/K$ is Galois, and its Galois group embeds in $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$, so it is abelian.

If $L$ is the splitting field of ${\Phi }_n$ over a characteristic-zero field $K$, choosing a primitive root ${\zeta }_n$ gives an injectionIt is surjective exactly when ${\Phi }_n$ is irreducible over $K$.

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 $n-\mathrm{deg}g$.