Learn English – How to use “their” and “theirs”

In grammatical terms, their is a possessive determiner, just like my, your, his, her, its and our. It indicates that what is described in the following noun phrase belongs to, or is in some other way associated with, the person or thing to which the possessive determiner refers. Just as my car is a car that belongs to me, so their house is a house that belongs to them.

There is a word with many uses. In your example, There are lots of skyscrapers in Dubai, it is what is sometimes called a ‘dummy subject’. It would be grammatical to say Lots of skyscrapers are in Dubai, but, generally, we don’t. We use there are (or there is) where French uses il est or il y a, German uses es ist, es sind or es gibt and Spanish uses hay.

First off, "in their field" has a different meaning from "in their fields".

field (n): a particular branch of study or sphere of activity or interest.

"Use" is a more generic verb that can be used in a wide array of situations, but it's fine in this sentence.

Farmers use chemical fertilizers in their fields.

"Apply" is more specific as it means "to put on the surface of", either literally or metaphorically. Also, the common verb + preposition is "apply to":

Farmers apply chemical fertilizers and insecticides to their fields.

While this is grammatical, it would be more idiomatic to use "spray on", because this is specifically how fertilizers and insecticides are applied.

Farmers spray chemical fertilizers and insecticides on their fields.