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The rank of a quadratic form is the rank of its associated symmetric bilinear form.

The factorizationshows that $X^3=-1$ always has $X=-1$ as a solution. For $p\ne 3$, it has two further solutions exactly when $-3$ is a quadratic residue; for $p=3$, all three factors coalesce at the unique root $-1$.

If $M/L$ is purely inseparable, an embedding of $L$ into an algebraic closure has at most one extension to $M$: if $x^{p^n}\in L$, the image of $x$ must be the unique $p^n$th root of the image of $x^{p^n}$. Existence in a normal overfield therefore implies uniqueness.

An algebraic element $x$ is purely inseparable over $K$ when $x^{p^n}\in K$ for some $n\ge 0$.

For a word consisting of three long constrained blocks, a pumpable window shorter than each block cannot alter both outer blocks; pumping then breaks a required three-way count relation.

A set of propositions has a model exactly when every finite subset has a model. Syntactically, a contradiction uses only finitely many assumptions, and every consistent set extends by Zorn lemma to a maximal consistent set in propositional logic.

A consistent set $T$ is maximal consistent when no proper extension is consistent. It is deductively closed and decides every formula: exactly one of $t$ and $\neg t$ belongs to $T$. Therefore $v(t)=1$ exactly when $t\in T$ defines a model of $T$.

Ifare deductively closed, then each $T_i$ is consistent and their union is consistent and deductively closed. The union is not finitary. Each conclusion follows from the finite character of formal proofs.

A set $S$ is finitary when it is deductively equivalent to some finite set $F$: both $S⊢F$ and $F⊢S$.

A set $S$ is syntactically consistent when $S/⊢\perp$. Equivalently in classical logic, there is no formula $\phi$ for which both $S⊢\phi$ and $S⊢\neg \phi$.

A set $S$ of formulae is deductively closed whenEquivalently, $S$ equals its deductive closure $\mathrm{Cn}(S)$.

If $\Gamma ⊢t$, every valuation that satisfies all formulae in $\Gamma$ also satisfies $t$.

This is soundness for the propositional proof system and valuation semantics.

Intuitionistic propositional logic omits unrestricted excluded middle and interprets implication constructively.

A uniformly accelerated observer has a null boundary beyond which light signals can never reach the observer.

A Möbius transformation is represented by an invertible two-by-two matrix, with nonzero scalar multiples representing the same map.

A Möbius transformation of the Riemann sphere has the formThe group is generated by translations, nonzero complex scalings, and inversion $z\mapsto 1/z$.

Over an algebraically closed field of characteristic other than two, the rank-$r+1$ quadricis an irreducible $r$-dimensional cone for $r\ge 2$, and its singular locus is the omitted-coordinate vertex $[0:\cdots :0:1]$.

If $X=V_{+}(F)\subseteq {\mathbb{P}}^n$, its affine cone is $Y=V(F)\subseteq {\mathbb{A}}^{n+1}$. If $X$ is smooth, every nonzero point of $Y$ is smooth; hence the vertex is the only possible singular point. The vertex is smooth for linear $F$ and singular whenever $\mathrm{deg}F\ge 2$.

Given homogeneous polynomials $F,G$ of the same degree, their pencil is the one-parameter familyIts total space is the projective hypersurface $V(sF+tG)\subseteq {\mathbb{P}}^2\times {\mathbb{P}}^1$, and projection to $[s:t]$ has these plane curves as its fibres.

It is a pencil whose members are plane curves.

For a projective algebraic set $X\subseteq {\mathbb{P}}^n$, its affine cone is the union in ${\mathbb{A}}^{n+1}$ of the lines represented by the points of $X$, together with the origin.