To adjoin an $n$th root of $g\in G$, let $m$ be the order of a group element $g$ and take a cyclic group $C=⟨a⟩$ of order $nm$ when $m$ is finite, or infinite order when $g$ has infinite order. In the free product $G\ast C$, $⟨a^n⟩$ and $⟨g⟩$ have the same order. The HNN extension with $t^{-1}{a}^{n}t=g$ therefore embeds its base by Britton's lemma, and $h=t^{-1}at$ satisfies $h^n=g$. If $g=1$, take $C$ of order $n$ and identify the two trivial subgroups. This is an existence construction, not an algorithm for finding an unknown element order.

In the HNN extension $⟨P,t\mid t^{-1}\ell t=\ell (\ell \in L)⟩$, the displayed equivalence follows from Britton's lemma. If $p\in L$, the word collapses by the defining isomorphism; otherwise it has stable letters and no pinch, so it is nonidentity. When $P$ is finitely presented and $L$ is finitely generated, imposing commutation with the finitely many subgroup generators gives a finite presentation. An undecidable membership problem for $L$ then becomes an undecidable word problem for a group in the extension.

Every finite-order element of an HNN extension is conjugate to a finite-order element of its base. Choose a conjugate minimizing the number of stable letters. If this number is positive, no pinch is possible either internally or across the end-to-start join, since a boundary pinch would allow a conjugation shortening it. Every positive power is then reduced with positive stable-letter length, and Britton's lemma makes it nonidentity. In particular, an HNN extension of a torsion-free group is torsion-free.

Near a flat quartic maximum, $V\simeq \alpha {\sigma }^4-3\alpha {\varphi }^4$. With reduced Planck mass $M$, the slow-roll approximation gives $N\simeq {\sigma }^4/(24M^2{\varphi }_N^2)$, ${\eta }_V\simeq -3/(2N)$ and ${ϵ}_V\simeq {\sigma }^4/(192M^4{N}^3)$. Hence $n_s\simeq 1-3/N$, $n_T\simeq -2{ϵ}_V$, and ${\mathcal{P}}_{\mathcal{R}}\simeq 8\alpha N^3/{\pi }^2$. The result is dominated by the near-maximum part of the trajectory; extending slow roll to a distant minimum is unnecessary.

// Target: cosmic-inflation.bigb

For a strongly continuous unitary representation of a compact group, this Bochner integral exists in the Hilbert-Schmidt norm and is a positive compact operator commuting with the representation. It is nonzero if $\xi \ne 0$, because $⟨K_{\xi }\xi ,\xi ⟩={\int }_G|⟨U_g\xi ,\xi ⟩{|}^{2}dg>0$. A nonzero eigenspace is finite-dimensional and invariant. Applying this on orthogonal complements yields an orthogonal direct sum of finite-dimensional irreducible representations, with no separability assumption on the represented Hilbert space.

Take $H_0=0$, $H_1={\sigma }_y/2$, ${\rho }_d=(I+{\sigma }_z)/2$ and $\rho =(I+{\sigma }_x)/2$. The overlap is $f=1/2$, but $C=\mathrm{Tr}({\rho }_d[-i{H}_1,\rho ])=-1/2$, so the Hilbert-Schmidt Lyapunov feedback identity gives $\dot{V}=1/2>0$. A target overlap and the derivative of that overlap along the control direction are different quantities.

For an unramified extension of non-Archimedean local fields, let $\varphi$ generate its cyclic Galois group. The ordinary Hilbert theorem 90 supplies $b\in L^{\times }$ with $x=\varphi (b)/b$. A common uniformizer $\pi \in K$ gives $y={\pi }^{-v_L(b)}b\in {\mathcal{O}}_L^{\times }$, retaining the same quotient because $\varphi$ fixes $\pi$. The reverse implication follows by telescoping the field norm.

For a Finite Galois extension $E/K(T)$, take a primitive element and exclude all ramification and denominator problems. A Hilbert subset keeping its minimal polynomial irreducible has residue degree $[E:K(T)]$. The residue group is an unramified decomposition subgroup of the generic Galois group with the same order, so it is the whole group. This specializes a regular Galois realization over $K(T)$ to realizations over $K$ when $K$ is Hilbertian.

A first-order embedded deformation of a closed subscheme $Z\subseteq X$ is a closed subscheme $Z_{\epsilon }\subseteq X{\times }_k\mathrm{Spec}(k[\epsilon ]/({\epsilon }^2))$ flat over the dual numbers and with special fibre equal to the given embedded $Z$. It is a tangent vector of the Hilbert scheme, not a deformation modulo arbitrary automorphisms of the ambient scheme. These deformations are naturally the global sections of the normal sheaf.

A finite abelian extension $E/K$ of number fields lies in the ordinary Hilbert class field exactly when it is unramified at every finite prime and every real place of $K$ splits completely in $E$. Its Galois group is then a quotient of the ordinary ideal class group through the Artin map.

This gives a useful recognition test: construct such an extension and show that its degree equals the class number $h_K$. The inclusion $E\subseteq H_K$ and equality of degrees then force $E=H_K$. Finite-prime unramifiedness alone gives the corresponding narrow-class-field condition when real places may become complex.

The ideal class group of $\mathbb{Q}(\sqrt{-30})$ is $C_2\times C_2$. Its four reduced binary quadratic forms have coefficients $(1,0,30)$, $(2,0,15)$, $(3,0,10)$ and $(5,0,6)$. The displayed multiquadratic field has degree four over the base. The seven quadratic-subfield discriminants are $8,-3,5,-24,40,-15,-120$, whose product in absolute value is $120^4$. The conductor-discriminant formula and discriminant tower formula imply that the relative discriminant over $\mathbb{Q}(\sqrt{-30})$ is the unit ideal. This unramified abelian degree-four extension is therefore its Hilbert class field.

The polynomial is irreducible with discriminant $229$, so its splitting field has Galois group $S_3$ and quadratic subfield $\mathbb{Q}(\sqrt{229})$. The splitting field is cyclic cubic over that subfield. It is unramified outside 229, and at 229 its inertia is a transposition, meeting the alternating cubic subgroup trivially. All roots are real. It is therefore everywhere unramified over the quadratic field. The ideal class group of Q of square root 229 has order three, so this extension is exactly the Hilbert class field.

The three reduced primitive binary quadratic forms of discriminant $-23$ are $(1,1,6)$ and $(2,\pm 1,3)$, so the class number is three. The displayed cubic has discriminant $-23$ and is irreducible. Its splitting field has Galois group $S_3$ and is cyclic cubic over $\mathbb{Q}(\sqrt{-23})$. Modulo $23$ it factors as $(X-10{)}^2(X-3)$. Lift the simple factor; the remaining quadratic has discriminant valuation one, so its local inertia group has order two. Its intersection with $A_3$ is trivial. Thus the relative cyclic cubic extension is everywhere unramified and equals the Hilbert class field.

For a CM-field $K$ over its maximal real subfield $K^{+}$, the ordinary Hilbert class field $H_{K^{+}}$ is totally real because its real places split. Thus $H_{K^{+}}\cap K=K^{+}$. Its compositum with $K$ is an abelian extension of $K$ of degree $h_{K^{+}}$, and is everywhere unramified by unramified extensions remain unramified under base change. It lies in $H_K$, giving the divisibility. The argument does not require injectivity of extension of ideal classes.

The bulk tangent covariance is twice the inverse Fourier transform of $k_{B}T/(A{q}^2+f)$, one term for each transverse direction. Equivalently the inverse of $-A{\partial }_r^2+f$ is $e^{-|r|/\xi }/(2\sqrt{Af})$. This gives the displayed correlation and identifies a force-dependent correlation length, different from the zero-force persistence length. Finite chains require endpoint conditions or a periodic convention.

Set $\epsilon =i{\tau }_2$, whose Pauli matrix identity is $\epsilon {\tau }_i^{\ast }=-{\tau }_i\epsilon$. Exponentiation gives $\epsilon U^{\ast }=U\epsilon$ for $U\in SU(2)$. Hence complex conjugating a Higgs doublet and multiplying by $\epsilon$ restores the fundamental representation of $SU(2)$, while reversing hypercharge. A Higgs field of hypercharge $1/2$ yields a conjugate doublet of hypercharge $-1/2$. Its product ${\tilde{\varphi }}^{†}L$ with a lepton doublet $L$ of hypercharge $-1/2$ transforms in the trivial representation of the whole electroweak gauge group. At vacuum $(0,v{)}^T/\sqrt{2}$, this composite is $v{\nu }_L/\sqrt{2}$.

A nonzero Higgs doublet has a real radial magnitude and three real angular coordinates. For a normalized vector $z=(z_1,z_2{)}^T$, the SU(2) matrix $U=\left(\begin{array}{cc}{z}_2^{\ast }& z_1\\ -z_1^{\ast }& z_2\end{array}\right)$ satisfies $U(0,1{)}^T=z$. Near the vacuum, the three broken generators can equivalently be parameterized by $e^{-i\mathbf{n}\cdot \mathbf{\sigma }+i{n}_3}$. A local gauge transformation removes these angular fields in unitary gauge. This coordinate choice is local about a nonzero vacuum and need not extend through zeros or topological defects.

Monolithic galaxy formation idealizes a galaxy as forming in one early rapid dissipative collapse, followed mainly by passive evolution. Rapid formation of the stars is distinct from early assembly of all those stars into their final object; hierarchical galaxy formation can assemble old stellar populations later.

Downsizing denotes the tendency for the characteristic mass of actively star-forming galaxies to decrease with time, together with related evidence that more massive quiescent systems formed their stars earlier or over shorter intervals. It does not assert that their dark-matter haloes assembled earlier: stellar formation and halo or stellar assembly are separate histories.

The galactic angular-momentum catastrophe is the excessive loss or redistribution of baryonic specific angular momentum in early simulated galaxy formation, producing overly compact disks or large central concentrations. Premature cooling, gravitational torques, interactions, and resolution contributed. Improved stellar feedback and gas accretion histories can mitigate it; it is a historical simulation problem rather than a theorem against cold dark matter.