The fundamental theorem of algebra factors $q(x,1)$ into linear factors over $\mathbb{C}$; homogenizing supplies factors of $y$ if the degree is less than three. Thus every nonzero homogeneous cubic is $c{L}_1{L}_2{L}_3$ for linear forms $L_i$, with $c\ne 0$ absorbed into one factor.

For pairwise nonproportional factors, part (c) assigns three distinct tangent lines ${\ell }_i=\mathbb{P}(L_i\cdot \mathrm{span}\{x,y\})$ to $C$. Their pairwise intersections are $[L_1{L}_2]$, $[L_1{L}_3]$, and $[L_2{L}_3]$. There cannot be a common point of all three: a nonzero quadratic cannot be divisible by three pairwise nonproportional linear factors, by unique factorization. Hence the three lines form a triangle circumscribed about $C$.

Assume $L\ne 0$. The quadratics divisible by $L$ form $L\cdot \mathrm{span}\{x,y\}$, a two-dimensional linear subspace; their projectivization is a projective line. Evaluating at the projective zero $[-\beta :\alpha ]$ of $L$ gives its equationAt the point $[L^2]=[{\alpha }^2:2\alpha \beta :{\beta }^2]$ of $C$, the tangent equation obtained from the gradient of $b^2-4ac$ is $-4{\beta }^{2}a+4\alpha \beta b-4{\alpha }^{2}c=0$. It is exactly the same line, establishing tangency.

Use coefficient coordinates $[a:b:c]$ on $\mathbb{P}(V)$. A nonzero binary quadratic has a repeated linear factor precisely when its discriminant vanishes, soIts gradient is $(-4c,2b,-4a)$, which cannot vanish at a projective point. Therefore $C$ is a nonsingular projective conic. The parametrization $[\alpha :\beta ]\mapsto [{\alpha }^2:2\alpha \beta :{\beta }^2]$ identifies it as a degree-two Veronese map.

A projective conic in ${\mathbb{P}}^2(\mathbb{C})$ is the zero locus of a nonzero homogeneous quadratic polynomial, up to nonzero scaling of that polynomial. Under an invertible linear change of homogeneous coordinates, its symmetric matrix has rank $3$, $2$, or $1$. Over $\mathbb{C}$, the corresponding forms are a nonsingular conic such as $XZ-Y^2=0$, two distinct lines such as $XY=0$, or a double line $X^2=0$. In the last case the repeated factor records multiplicity, even though the underlying point set is one line.

A projective line is the projectivization of a two-dimensional linear subspace of ${\mathbb{C}}^3$, equivalently the zero locus of one nonzero linear form. For two such subspaces $U,W$, the dimension formula for a sum of subspaces gives $\dim (U\cap W)\ge 2+2-3=1$. A nonzero vector in the intersection therefore supplies a common projective point. Distinct lines have exactly one such point.

Suppose $N$ were a normal subgroup of index $m$ and put $Q=SL(2,{\mathbb{F}}_p)/N$. The supplied reduction map and part (c) show that $Q$ is generated by the images $s,t$ of $S,ST$. Their orders divide $4$ and $6$, respectively, because $S^4=I$ and $(ST{)}^6=I$. By Lagrange's theorem, their orders also divide $|Q|=m$. Since $\gcd (m,6)=1$, both $\gcd (m,4)$ and $\gcd (m,6)$ equal one. Thus $s=t=1$, forcing $Q$ to be trivial and $m=1$, a contradiction. No such normal subgroup exists for $m>1$.

Multiplication, followed by induction and inversion for negative powers, givesIf $c\ne 0$, apply Euclidean division to write $a=q|c|+r$ with $0\le r<|c|$. Choose $n=-q\mathrm{sgn}(c)$, so $a+nc=r$. This is one step of the Euclidean algorithm implemented by a determinant-one row operation.

A group presentation $⟨X\mid R⟩$ denotes the quotient $F(X)/⟨⟨R⟩⟩$ of the free group on the generator symbols by the normal closure of the relator words. The symbols generate the quotient, and each relator becomes the identity. The phrase also asserts that no additional relations are imposed except consequences of these relators and the group axioms. Equivalently, an assignment of the generators to a group extends to a group homomorphism from the presented group exactly when all relators evaluate to the identity; this is the universal property of a group presentation.