ExtendField => bVarious functions which produce points, or call functions which produce points, may naturally find scheme theoretic points that are not rational over the base field (for example, by intersecting with a random linear space). Setting ExtendField => true will tell the function that such points are valid. Setting ExtendField => false will tell the function ignore such points. This sometimes can slow computation, and other times can speed it up. In some cases, points over extended fields may also have better randomness properties for applications.
The object ExtendField is a symbol.