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Let $g\in K[T]$ in characteristic $p$, let $f(T)=g(T^p)$, let $L$ split $g$ over $K$, and let $M$ split $f$ over $L$. If $y_i$ are the roots of $g$ and $x_i^p=y_i$, then the Frobenius endomorphism makes each $x_i$ unique andbecause $y_i=x_i^p$. Thus $M$ is already the splitting field of $f$ over $K$, and every root of $f$ is purely inseparable over $L$.

A finite extension $L/K$ is normal exactly when it is the splitting field over $K$ of some polynomial. For the forward direction, multiply the minimal polynomials of finitely many generators of $L$; normality puts all their roots in $L$. The reverse direction follows because a $K$-embedding permutes the roots of a polynomial and hence preserves its splitting field.

If $\alpha =\sqrt[3]{2}$ is the positive real root and ${\zeta }_3=e^{2\pi i/3}$, then the roots of $X^3-2$ areTheir splitting field over $\mathbb{Q}$ is $\mathbb{Q}(\alpha ,{\zeta }_3)$.

Every nonconstant polynomial $f\in K[X]$ has a splitting field of finite degree over $K$. It is constructed by adjoining a root of an irreducible factor and repeating over the enlarged field. Any two splitting fields of $f$ over $K$ are isomorphic by an isomorphism that fixes $K$. Inside one fixed algebraic closure, the field generated by all roots is unique as a subfield.

A spin coherent state is obtained by rotating the maximal-weight state $|s,s⟩$ so that it is an eigenstate of spin along a chosen direction with eigenvalue $s\hslash$.

In descending $S_z$ order, the spin-$3/2$ raising operator has superdiagonal entries $\hslash \sqrt{3},2\hslash ,\hslash \sqrt{3}$; $S_x$ and $S_y$ are its Hermitian real and imaginary combinations.

The operators $S_{\pm }=S_x\pm i{S}_y$ satisfy

The spin-$s$ irreducible representation has dimension $2s+1$, with basis $|s,m⟩$ for $m=-s,-s+1,\dots ,s$ and Casimir eigenvalue ${\hslash }^{2}s(s+1)$.

For $\mathbf{n}=(\sin \theta ,0,\cos \theta )$, the normalized eigenstates of $\mathbf{n}\cdot \mathbf{S}$ arewith eigenvalues $+\hslash /2$ and $-\hslash /2$.

For $N$ noninteracting units with energies $-ϵ,0,ϵ$, put $x=\beta ϵ$. Thenand

A vacuum cavity of radius $a(t)$ in an infinite inviscid incompressible fluid of density $\rho$ and far-field pressure $p_0$ obeysIf $a(0)=a_0$ and $\dot{a}(0)=0$, its collapsing branch satisfies

The convex hull of finitely many unit-circle points contains zero exactly when the points do not lie in any open semicircle, equivalently when their shortest containing arc has length at least $\pi$.

For $U_{\gamma }=\mathrm{diag}(1,e^{i\gamma })$ with $0\le \gamma <2\pi$,

A Laplacian term in the infected-population equation models unbiased random migration of infective individuals.

Mass-action transmission at unit nondimensional rate transfers population from susceptible to infected compartments at rate $SI$.

A directed spanning tree rooted towards a vertex $r$ is a spanning tree whose edges are oriented so that every other vertex has a unique directed path to $r$.

A uniform spanning tree is sampled uniformly from all spanning trees of a finite connected graph.

This is the sequence space of sequences with a finite limit.

The space $c$ consists of convergent scalar sequences with the supremum norm.

If $\epsilon :\mathbb{R}\to [0,\infty )$ satisfies $\epsilon (t)\le M|t|$, thenThus these square-root perturbations converge uniformly to $|x|$.

With the supremum norm, $C_0({\mathbb{R}}^d)$ is complete. A uniform Cauchy sequence has a uniform continuous limit, and one member controls the limit uniformly outside a large ball. Every member is also uniformly continuous: use uniform continuity on a large compact ball and smallness of the function beyond it.