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An imaginary-time-ordered correlation function orders operators along the Euclidean thermal circle. Bosonic correlators are periodic with period $\beta$.

For a bosonic operator $A$, the retarded Green function is $G_R(t)=-i\theta (t)⟨[A(t),A(0)]⟩$. Causality makes its Fourier transform analytic in the upper half frequency plane.

The nonzero elements of every finite field form a cyclic group. For a field with $q$ elements, combine elements realizing each prime-power factor of the group exponent to obtain an element of exponent order; the root bound for $X^m-1$ then forces that order to be $q-1$.

Primordial non-Gaussianity is any departure of the primordial perturbations from Gaussian random fields. Cubic interactions generate a nonzero three-point correlation function at leading order.

This is non-Gaussianity of primordial cosmological perturbations rather than non-Gaussian statistics of an arbitrary physical measurement.

A $P(X,\varphi )$ scalar theory has Lagrangian density $P$ depending on the field and $X=-g^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi /2$. It includes canonical scalar theory and models with noncanonical kinetic interactions.

A k-essence model uses noncanonical kinetic terms for cosmological acceleration; a general P(X, phi) scalar field theory need not have that application.

If $L/K$ and $K/F$ are algebraic field extensions, then $L/F$ is algebraic. In particular, an element algebraic over an algebraic extension of $F$ is algebraic over $F$.

This is a closure property of algebraic extensions, not an alternative name for an extension.

A quadratic extension is a field extension of degree two. It is normal because the other root of the minimal polynomial $T^2+bT+c$ of any non-base-field element is $-b-\alpha$ and lies in the same extension.

For a field extension $L/K$, an intermediate field is a field $M$ satisfying $K\subseteq M\subseteq L$.

An algebraic closure of a field $K$ is an algebraic extension $\overline{K}/K$ for which $\overline{K}$ is algebraically closed.

Inside a common overfield, the compositum $L_1{L}_2$ is the smallest field containing both $L_1$ and $L_2$.

The multiplication image of the tensor product of fields in a common overfield generates the compositum as a field. For algebraic extensions that image is already a field; in general it need not be. The compositum is therefore distinct from the tensor product algebra.

For finite extensions $K\subseteq L\subseteq M$,

The characteristic of a field $F$ is the least positive integer $n$ for which $n\cdot 1_F=0$, or zero if no such integer exists.

A field is algebraically closed when every nonconstant polynomial over it has a root in it, equivalently when every such polynomial splits into linear factors.

The lowering operator sends the occupied state to the vacuum and kills the vacuum; its adjoint performs the reverse and kills the occupied state.

The fermionic occupation Hamiltonian satisfies $H^2=H$, so its spectrum lies in $\{0,1\}$.

Write the even number $N-1$ as $2^{s}m$ with $m$ odd. An odd composite number $N$ is a strong pseudoprime to a coprime base $b$ whenor $b^{2^{j}m}\equiv -1(\mathrm{mod}N)$ for some $0\le j<s$.

For fixed $a>1$ and suitable odd primes $p$, the composite generalized repunitis congruent to one modulo $2p$ and is a pseudoprime to base $a$.

For nested Gaussian linear models differing by $r$ parameters, the statistichas an $F_{r,n-p_1}$ distribution under the reduced model.

The squared Hellinger distance iswhere $p$ and $q$ are densities with respect to any common dominating measure $\lambda$.

For a measurable function $T$, every f-divergence contracts under pushforward:This follows from conditional Jensen inequality applied to the jointly convex perspective $(a,b)\mapsto bf(a/b)$.