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Let $Q$ be the subgroup of nonzero squares in ${\mathbb{F}}_{11}^{\times }$. The affine mapsform the nonabelian semidirect product $C_{11}⋊C_5$, with multiplication

For an odd prime $p$, the splitting fieldof $X^p-2$ has Galois groupThe translation subgroup multiplies $2^{1/p}$ by powers of ${\zeta }_p$, while the multiplicative subgroup acts on ${\zeta }_p$ by cyclotomic automorphisms.

The algebraic sethas the two irreducible componentsEach is irreducible because the linear change $u=x-z$, $v=x+z$ identifies its coordinate ring withan integral domain.

For ideals $I,J\subseteq k[x_1,\dots ,x_n]$,The second equality also follows from $V(IJ)=V(I)\cup V(J)$ and

For an affine algebraic set $X\subseteq {\mathbb{A}}_k^n$, its coordinate ring isThe set $X$ is irreducible exactly when $k[X]$ is an integral domain.

For an algebraic group $H$, a flat $H$-torsor over $Y$ is a faithfully flat morphism $X\to Y$ with a right $H$-action such that $X\times H\to X{\times }_{Y}X$, $(x,h)\mapsto (x,xh)$, is an isomorphism. A Zariski torsor additionally becomes $U\times H$ on a Zariski open cover of $Y$.

A unipotent algebraic group is built from unipotent elements. A unipotent algebraic group is an affine algebraic group all of whose elements are unipotent. The Kolchin theorem embeds it into an upper unitriangular group.

The derived subgroup $[G,G]$ is the closed algebraic subgroup generated by the commutators $xy{x}^{-1}{y}^{-1}$. If $G$ is connected, images of finite products of commutators are connected and their closures stabilize by dimension, which shows that $[G,G]$ is connected.

This is the multiplicative Jordan–Chevalley decomposition in a linear algebraic group. Every element $g$ of an affine algebraic group has unique commuting semisimple and unipotent parts $g_s,g_u$ such that $g=g_s{g}_u$. The construction agrees with the multiplicative Jordan decomposition in every rational representation.

An element of an affine algebraic group is unipotent when its image in one, equivalently every, faithful finite-dimensional representation has every eigenvalue equal to one.

An element of an affine algebraic group is semisimple when its image in one, equivalently every, faithful finite-dimensional representation is a diagonalizable linear map.

The group $G\times G$ acts on $k[G]$ byFor $G={\mathbb{G}}_m$, this gives $k[G]={⨁}_{r\in \mathbb{Z}}{k}_{(-r,r)}$. For $G={\mathbb{G}}_a$, the action factors through $(a,b)\mapsto b-a$ and the degree filtration of $k[t]$ has trivial one-dimensional successive quotients.

Every affine algebraic group has a faithful finite-dimensional rational group representation. Choose algebra generators of $k[G]$ and place them in a finite-dimensional subspace stable under right translations. An element acting trivially on that subspace has the same coordinate values as the identity and therefore is the identity.

The Cauchy problem with initial value $u_0$ has

A category with finite products is cartesian closed when every product functor $-\times A$ has a right adjoint $(-{)}^A$, called exponentiation by $A$.

The colimit form of the Special adjoint functor theorem says that a colimit-preserving functor from a locally small, cocomplete, well-copowered category with a small generating family to a locally small category has a right adjoint.

The right Kan extension along $K:\mathcal{C}\to \mathcal{D}$ is the right adjoint to precomposition by $K$ between suitable functor categories.

This is the right-adjoint form of Kan extension; left Kan extension is its left-adjoint counterpart.

For $F⊣G$, the unit $\eta :1\to GF$ and counit $\epsilon :FG\to 1$ are the natural transformations corresponding to identity morphisms under the adjunction. They satisfy ${\epsilon }_{F}F\eta =1_F$ and $G\epsilon {\eta }_G=1_G$.

For a fixed amount of ideal gas with constant heat capacities, the first law and $\delta Q=0$ givewhere $\gamma =C_p/C_V$.

A dusty radiation-pressure-driven shell is swept up when ultraviolet radiation from an active nucleus is absorbed by dust and reradiated in the infrared. Single scattering supplies up to roughly $L/c$ of momentum flux, while trapped infrared photons can enhance it by the infrared optical depth.