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In a vector space over a field $k$, an affine line is a translate of a one-dimensional vector subspace. Its direction is that vector subspace, so multiplying $v$ by a nonzero scalar preserves the direction. Over a finite field of size $q$ it contains exactly $q$ distinct points. This usage concerns affine subsets of vector spaces, rather than the affine line as an affine scheme.

A $C_2$ field is a Ci field with the bound $N>d^2$. Function fields of curves over finite fields are examples. The extra-variable construction in surjectivity of reduced norms over C2 fields makes this polynomial-solvability property useful for division algebras.

A field is $C_i$ if every homogeneous polynomial of degree $d\ge 1$ in more than $d^i$ variables has a nonzero zero. Algebraically closed fields are $C_0$, finite fields are $C_1$, and the Lang-Nagata theorem for Ci fields controls function-field extensions.

The vector space of formal polynomials over a field $k$ in $n$ variables with total degree of a polynomial at most $d$ has the monomial basis $x_1^{a_1}\cdots x_n^{a_n}$ for nonnegative exponents with ${\sum }_i{a}_i\le d$. Introducing a slack exponent and using stars and bars counts $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ basis elements. These are formal polynomials; over a finite field, distinct formal polynomials need not define distinct polynomial functions unless suitable degree restrictions hold.

The Fano plane is the projective plane over the finite field ${\mathbb{F}}_2$. It has seven points and seven lines, each line contains three points, and two distinct lines meet in one point. A line-point incidence matrix $N$ therefore satisfies $N{N}^{\mathsf{T}}=2I+J$, making it invertible over the real numbers and useful for the transplantation theorem.

The finite field ${\mathbb{F}}_9$ can be represented as ${\mathbb{F}}_3[X]/(X^2+1)$, since the quadratic has no root modulo three. Its elements are $a+b\beta$ for $a,b\in {\mathbb{F}}_3$, with ${\beta }^2=-1$. Multiplication reduces to $(a+b\beta )(c+d\beta )=(ac-bd)+(ad+bc)\beta$ modulo three.

For a nonconstant polynomial over any field, use triangular substitutions $x_i=y_i+y_n^{B^i}$, $x_n=y_n$, where $B$ exceeds every exponent in its support. Base-$B$ weights distinguish all original monomials, so the transformed polynomial has a unique highest $y_n$ power with coefficient in $k^{\times }$. Rescale to make it monic. The monic polynomial quotient is finite free over $k[y_1,\dots ,y_{n-1}]$, yielding a finite flat projection. The nonlinear substitution works over finite fields as well as infinite ones; a generic linear-direction argument alone would not do so.

Upper-unitriangular three-by-three matrices over the prime finite field form the finite analogue of the real Heisenberg group. In coordinates their multiplication is $(a,b,c)(a^{\prime },b^{\prime },c^{\prime })=(a+a^{\prime },b+b^{\prime },c+c^{\prime }+a{b}^{\prime })$. The order is $p^3$, and $(a,b,c{)}^k=(ka,kb,kc+\left(\genfrac{}{}{0ex}{}{k}{2}\right)ab)$. For odd $p$, every nonidentity element has order $p$, but the group is nonabelian. For $p=2$ the exponent statement changes, so odd characteristic is essential for the nonisomorphic Gassmann equivalent regular subgroups construction.

Elliptic curves isogenous over a finite field have the same number of rational points: the isogeny intertwines Frobenius, and multiplicativity of degree gives equal degrees for $1-\pi$. Their groups need not be isomorphic. Over ${\mathbb{F}}_7$, $y^2=x^3-x$ and $y^2=x^3+4x$ are two-isogenous and have groups $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/8\mathbb{Z}$, respectively.

Over a finite field of odd cardinality $q$, a diagonal nondegenerate ternary quadratic form $A{x}^2+B{y}^2+C{z}^2$ has a zero with $z=1$: the sets $\{A{x}^2\}$ and $\{-C-B{y}^2\}$ both have $(q+1)/2$ elements and must intersect. This gives a nonzero nonsingular zero. Over a local integer ring with these coefficients units, the Hensel lemma lifts it to an isotropic vector in the fraction field.

For any field $F$ and vector subspace $W\subseteq F^m$, some $u\in W$ has at least $\dim W$ nonzero coordinates. Choose $u$ with largest support of a vector $S$. Restriction from $W$ to $F^S$ is injective: a nonzero vector in its kernel of a linear map vanishes on $S$, so adding it to $u$ would enlarge $S$. Thus $\dim W\le |S|$. The argument works over finite fields, where assuming a generic vector avoids finitely many hyperplanes would be invalid.

The modular Heisenberg group consists of upper-unitriangular $3\times 3$ matrices over the finite field ${\mathbb{F}}_p$. In coordinates $M(x,y,z)$, multiplication is $M(x,y,z)M(x\prime ,y\prime ,z\prime )=M(x+x\prime ,y+y\prime +xz\prime ,z+z\prime )$, and its order is $p^3$. Its center of a group is $\{M(0,y,0)\}\cong C_p$; the quotient group by the centre is $C_p\times C_p$, represented by the $x,z$ coordinates. The elementary generators $a=M(1,0,0)$, $b=M(0,0,1)$, $c=M(0,1,0)$ satisfy $a^p=b^p=c^p=e$, $ab{a}^{-1}{b}^{-1}=c$, and $c$ central. For $p=2$, $(ab{)}^2=c$, so the whole group does not have exponent two.

Over the reals or a finite field with more than two elements, the normalizer of all invertible diagonal matrices in $G{L}_2$ consists of monomial matrices and has quotient $C_2$ over that diagonal subgroup. Over ${\mathbb{F}}_p$ its size is $2(p-1{)}^2$ for $p>2$. For $p=2$ the diagonal subgroup is trivial, so its normalizer is all of $G{L}_2({\mathbb{F}}_2)$, of order six.

For prime $p$, every order-$p$ element of $G{L}_2({\mathbb{F}}_p)$ is conjugate to the displayed nonidentity unipotent matrix. Its cyclic subgroup has vector group orbits of sizes one or $p$; counting all $p^2$ vectors implies at least $p$ fixed vectors. Choose a nonzero fixed vector as first basis vector. The matrix becomes triangular with first diagonal entry one; its other diagonal entry satisfies $d^p=1$ and hence $d=1$ in the prime finite field. The off-diagonal entry is nonzero and can be rescaled to one. All matrices involved have invertible changes of basis.