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A simple closed curve of length $L$ on the unit sphere enclosing the smaller area $A$ satisfiesEquality holds for a circle. A variational proof uses Gauss--Bonnet to turn area maximization at fixed length into geodesic-curvature minimization, proves that an extremizer has constant geodesic curvature and is planar, and evaluates its spherical cap.

In the curvature-minus-one Poincaré half-plane model, a geodesic polygon with $n$ sides and interior angles ${\alpha }_1,\dots ,{\alpha }_n$ has areaThis is the polygonal Gauss-Bonnet theorem.

Gauge fixing removes the degeneracy among gauge-equivalent field configurations so that the kinetic operator has an inverse propagator.

Yang-Mills theory is the non-Abelian gauge theory with field strengthUnder an infinitesimal gauge transformation it transforms covariantly as $\delta F_{\mu \nu }=-ig[F_{\mu \nu },\alpha ]$.

At an upper critical dimension, a nominal power-law critical scaling acquires a multiplicative logarithmic correction because the coupling is marginal.

The functional equation and $\Gamma (1)=1$ give $\Gamma (z)\sim 1/z$ as $z\to 0$ away from the negative real axis.

For $\mathrm{Re}p,\mathrm{Re}q>0$, Euler's beta function is

It follows by making the logarithm of the quotient entire and periodic; growth forces its periodic remainder to be constant.

The digamma function is the logarithmic derivativeLogarithmically differentiating the Weierstrass product gives

The reciprocal gamma function has the entire-product representation

This is a particular application of the Weierstrass factorization theorem.

For $\mathrm{Re}s>1$, expansion of $(e^x-1{)}^{-1}$ as a geometric series and termwise integration give

The Gamma function obeys the recurrence relationInitially, for $\mathrm{Re}z>0$, integration by parts in the Gamma integral gives the short proofThe boundary term vanishes at zero because $|t^z|=t^{\mathrm{Re}z}\to 0$, and at infinity because of exponential decay. Together with $\Gamma (1)={\int }_0^{\infty }{e}^{-t}dt=1$, the Gamma function recurrence immediately gives $\Gamma (n+1)=n!$ for every nonnegative integer $n$.^[ref]

Read backwards, the Gamma function recurrence defineswhere $m$ is a positive integer. These overlapping formulas agree by the Gamma function recurrence, giving the meromorphic continuation. Since $\Gamma (1)=1$, they give a simple pole at zero and then at every negative integer; Gamma function residue at a nonpositive integer records the residues. The recurrence relation therefore holds throughout the continuation as an identity of meromorphic functions.

The discriminant square class detects whether the transitive cubic Galois group lies in the alternating group.

The polynomial $X^3-X-1$ is irreducible over $\mathbb{Q}$ and has discriminant $-23$, so its splitting field has Galois group $S_3$. Its unique quadratic subfield is the fixed field of $A_3$, namely $\mathbb{Q}(\sqrt{-23})$.

Suppose a group of field automorphisms acts transitively on elements $y_i$, and each $y_i$ has a unique $p$th root $x_i$ in a stable extension. If every automorphism extends, it sends $x_i$ to the unique root above its image of $y_i$, so the extended group acts transitively on the $x_i$.

Put $\alpha =\sqrt[4]{7}$ and $L=\mathbb{Q}(\alpha ,i)$. The dihedral group $D_8=⟨r,s:r^4=s^2=1,srs=r^{-1}⟩$ acts byThe ten subgroups and their fixed fields areThe normal subgroups are $D_8$, $⟨r⟩$, the two displayed Klein four-groups, $⟨r^2⟩$, and $\{1\}$.

Every subgroup of an abelian group is normal. Hence every intermediate field of a finite abelian Galois extension is itself Galois over the base field.

For a finite Galois extension $L/K$ with group $G$, an intermediate extension $E/K$ is normal exactly when its corresponding subgroup $\mathrm{Gal}(L/E)$ is a normal subgroup of $G$.

The disk--halo degeneracy is the ability of different stellar mass-to-light ratios and dark-halo profiles to reproduce nearly the same total galaxy rotation curve.