Vector space over a finite field

If ${\mathbb{F}}_q$ is a finite field, a vector space of finite dimension $d$ over ${\mathbb{F}}_q$ has $q^d$ elements. Every infinite-dimensional vector space over ${\mathbb{F}}_p$ is an infinite elementary abelian $p$-group under addition.

For infinite spaces over a field of size $q$, two-variable types are determined by the kernel of the linear map sending $(a,b)$ to $ax+by$. Its ranks give the zero pair, $q+1$ one-dimensional directions and the independent-pair type. All $q+3$ types are isolated by finite formulas. Quantifier elimination makes the Lindenbaum algebra their full power-set Boolean algebra. The one-variable algebra has just the zero and nonzero atoms, hence four Elements.

A subset of ${\mathbb{F}}_q^n$ is a finite-field Kakeya set when it contains an affine line in a vector space in every one-dimensional direction. The line position may depend on the direction. The polynomial method in combinatorics yields the finite-field Kakeya polynomial bound on its cardinality.

Work in the affine plane over a finite field ${\mathbb{F}}_q$ of odd characteristic. The union of the affine lines $y=mx-m^2/4$, for $m\in {\mathbb{F}}_q$, contains a line of every finite slope. Completing the square givesFor a fixed $x$, the union therefore contains exactly the $(q+1)/2$ values of $y$ for which $x^2-y$ is a square, including zero. Adding the vertical affine line $x=0$ completes all directions. The resulting finite-field Kakeya set has cardinalityIts density tends to one half as $q$ grows through odd prime powers. Geometrically the finite-slope lines are the tangents to the parabola $y=x^2$, because the tangent at $x=m/2$ has the displayed equation.

For a finite-field Kakeya set $A\subseteq {\mathbb{F}}_q^n$, a smaller cardinality would give a nonzero polynomial of total degree of a polynomial at most $q-1$ vanishing on $A$, by the dimension of a bounded-total-degree polynomial space. Its restriction to each complete affine line in a vector space has $q$ roots of a polynomial and degree less than $q$, hence vanishes identically. The top homogeneous polynomial part then vanishes in every direction. A polynomial of degree less than $q$ in each variable cannot vanish everywhere on ${\mathbb{F}}_q^n$ unless it is zero, giving a contradiction. For the prime field, this is the polynomial nonvanishing below the field size.