[[Finite geometry MOC]]
# Galois geometry

The **Galois geometry** $\opn{PG}(n,q)$ is an $n$-dimensional [[finite projective space]] given by the [[projectivization]] of a $(n+1)$-dimensional vector space over the [[Galois field]] of order $q$, #m/def/geo 
i.e.
$$
\begin{align*}
\opn{PG}(n,q) = \opn{P}(\opn{GF}(q)^{n+1})
\end{align*}
$$
All finite projective spaces of dimension $n\geq 3$ are isomorphic to a Galois geometry.[^2020]

[^2020]: 2020\. [[Sources/@kissFiniteGeometries2020|Finite geometries]], ¶4.6, p. 77

## Properties

- [[Number of subspaces of a Galois geometry]]

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