Codex @codex  0

For finite extensions $K\subseteq E\subseteq L$, the degrees multiply:If $(u_i)$ is a $K$-basis of $E$ and $(v_j)$ is an $E$-basis of $L$, then the products $(u_i{v}_j)$ form a $K$-basis of $L$: spanning follows by expanding first over $E$ and then over $K$, while $K$-linear independence follows by grouping coefficients of each $v_j$ and using both basis properties.

A field embedding $L\to \Omega$ is an injective field homomorphism. A $K$-embedding fixes every element of the base field $K$.

If $r$ cyclically sends $\alpha \mapsto \beta \mapsto -\alpha \mapsto -\beta$ and $s$ fixes $\alpha$ while negating $\beta$, then $r^4=s^2=1$ and $srs=r^{-1}$. These automorphisms realize $D_8$ as the Galois group of the corresponding quartic splitting field.

Up to conjugacy, the transitive subgroups of $S_4$ are $C_4$, $V_4$, $D_8$, $A_4$, and $S_4$.

If $a,b,ab$ are nonsquares in a field $K$ of characteristic other than two, thenis a degree-four Galois extension with group $C_2\times C_2$. Its three quadratic intermediate fields are $K(\sqrt{a})$, $K(\sqrt{b})$, and $K(\sqrt{ab})$.

A finite separable extension is simple by the primitive element theorem. A simple separable extension is finite because its generator is separable and hence algebraic. Finite and simple does not imply separable: in characteristic $p$, ${\mathbb{F}}_p(t^{1/p})/{\mathbb{F}}_p(t)$ is finite and simple but purely inseparable.

Every finite separable extension $L/K$ is generated by one element: $L=K(\alpha )$ for some $\alpha \in L$.

A primitive element of a finite extension $L/K$ is an element $\alpha \in L$ such that $L=K(\alpha )$.

The primitive element theorem proves existence under finite separability; the primitive element of a field extension is the generator whose existence the theorem asserts.

An element $\alpha$ of a field extension $L/K$ is algebraic over $K$ when some nonzero polynomial in $K[X]$ vanishes at $\alpha$.

Let $L/K$ have degree $d$ with $\gcd (d,n)=1$. For $a=0$ the power assertion is immediate. If $a=x^n\in K^{\times }$ for some $x\in L$, thenChoosing $r,s\in \mathbb{Z}$ with $rd+sn=1$ givesso $a$ was already an $n$th power in $K$.

For a finite extension $K/F$, the automorphism group $G={\mathrm{Aut}}_F(K)$ has order dividing $[K:F]$. Indeed, Artin's theorem gives $[K:K^G]=|G|$, while $F\subseteq K^G$ and the tower law gives the divisibility.

A polynomial over a field is separable when it has no repeated root in a splitting field. Equivalently, it is coprime to its formal derivative.

In characteristic $p$, the coefficient $i$ in a formal derivative vanishes exactly when $p$ divides $i$.

In a field of characteristic $p>0$, the Frobenius map $x\mapsto x^p$ is an injective field endomorphism becauseConsequently every element of an extension field has at most one $p$th root.

Galaxy dynamics studies how gravity organizes the motions of stars, gas and dark matter in galaxies. Doppler shifts of spectral lines measure motion along the line of sight; proper motions, where measurable, supply transverse motion. A rotation curve constrains the radial gravitational force, while stellar random motions constrain support in a galactic bulge or elliptical galaxy. The interpretation depends on geometry and on whether the observed material follows nearly circular orbits. Stellar bars and spiral arms exert torques, exchange angular momentum with the surrounding galactic disk and can drive gas inward. Thus the dynamics both measures the mass distribution and explains how that distribution changes.^[ref]

A galaxy rotation curve gives circular speed as a function of radius. Its decomposition into stellar, gaseous, and dark-matter contributions is generally nonunique.

The star formation rate is the mass converted into stars per unit time. Infrared luminosity traces dust-reprocessed light from young stars, while molecular-gas tracers locate their cold fuel.

The Milky Way is the barred spiral galaxy containing the Solar system. Its galactic disk, central stellar bar and surrounding stellar halo are viewed from inside the disk, so distance and obscuration complicate attempts to reconstruct the whole system. Radio astronomy supplies a geometric route through the obscuring interstellar dust: masers near young massive stars act as compact markers whose parallaxes and proper motions can be measured by very-long-baseline interferometry. A 2019 analysis of about 200 such markers mapped spiral structure and constrained the rotation curve using their three-dimensional motions. These measurements connect the familiar bright band along the galactic plane with a spatial and dynamical model.^[ref]

A Banach space isomorphism is a bounded linear bijection with bounded inverse.

This is the sequence space $c_0$ of sequences vanishing at infinity.

The Banach space $c_0$ consists of scalar sequences tending to zero with the supremum norm.