What is the meaning of the word "field"?

Definitions:

(mathematics) a set of elements such that addition and multiplication are commutative and associative and multiplication is distributive over addition and there are two elements 0 and 1 noun
- the set of all rational numbers is a field
a geographic region (land or sea) under which something valuable is found noun
- the diamond fields of South Africa
a particular kind of commercial enterprise noun
- they are outstanding in their field
a piece of land cleared of trees and usually enclosed noun
- The farmer planted a field of wheat.
a piece of land prepared for playing a game noun
- the home crowd cheered when Princeton took the field
a region in which active military operations are in progress noun
- the army was in the field awaiting action
a region where a battle is being (or has been) fought noun
- they made a tour of Civil War battlefields
answer adequately or successfully verb
- The lawyer fielded all questions from the press
select (a team or individual player) for a game verb
- The Buckeyes fielded a young new quarterback for the Rose Bowl
somewhere (away from a studio or office or library or laboratory) where practical work is done or data is collected noun
- anthropologists do much of their work in the field