Re: what is a field?
"Field" is a concept in "Abstract Algebra". Simply put, a "field" is such a set that satisfies:
(1) it has a "zero" and a "unit";
(2) it is closed with respect to addition, subtraction, multiplication and division (except by "zero").
Example 1. The set of all rational numbers is a field.
Example 2. The set of all real numbers is a field.
Example 3. The set of all complex numbers is a field.
Example 4. The set of all nonnegative rational numbers is NOT a field, because it is not closed with respect to subtraction. For instance, 1-2 = -1 which is not in the set.
Example 5. The set of all integers is NOT a field because it is not closed with respect to division. For instance, 1/4=0.25 - not an integer.
Example 6. The set of all matrices is NOT a field because there exist such matrices which are not zero but don't have inverses (not closed with respect to division).
In your case, the entries are assumed to be in a field. I believe it emphasizes that you don't have to use real numbers as the components as usual, and you may want to use other numbers like rational numbers, complex numbers, etc.