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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 $\gcd (d,n)=1$. If $a=x^n\in K$ 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 the gravitational motions and evolution of stars, gas, dark matter, and satellite systems in galaxies.

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.

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.

For conjugate exponents $p,q$,with the analogous inequality ${\sum }_n|x_n{y}_n|\le \Vert x{\Vert }_p\Vert y{\Vert }_q$ for sequences.

The continuous dual space $X^{\ast }$ of a topological vector space is the vector space of its continuous linear functionals. Unlike the algebraic dual, it requires that a small change in an input produce a small change in its scalar measurement. For a normed vector space, this is equivalent to a finite boundfor all vectors $x$ and some constant $C\ge 0$. The least such constant is the operator norm of $\phi$.

This is the continuity-restricted subspace of the algebraic dual space; in infinite dimension the two can differ.

For the linear functional $\phi (x,y)=2x-y$ on ${\mathbb{R}}^2$ with its Euclidean norm, the Cauchy-Schwarz inequality givesEquality holds on the unit vector $(2,-1)/\sqrt{5}$, so its operator norm is exactly $\sqrt{5}$. The same estimate controls measurement error: $|\phi (v+h)-\phi (v)|\le \sqrt{5}\Vert h{\Vert }_2$. The linear functional diagram shows which changes are entirely invisible to this measurement; its kernel of a linear map consists of the changes with zero response. All linear functionals on a finite-dimensional real or complex normed vector space are continuous, but discontinuous functional detected by vanishing sequence tails shows why the restriction matters in infinite dimension.

Translations of a compactly supported bump can form an infinite family separated by a fixed sup-norm distance, disproving compactness.

C0(X) consists of continuous functions whose values become arbitrarily small outside compact sets.