"The farmer has many fields."

Translation: Ο αγρότης έχει πολλά χωράφια.

[chrysaphi]

...and if it's a female farmer?

[D_..]

Η αγρότισσα.

[HellasCad]

Why is ο γεωργός wrong?

[shivys13]

"Πολλοί κάμποι" wouldn't be another option?

[D_..]

No, unfortunately not. Κάμπος (http://www.wordreference.com/gren/%CE%BA%CE%AC%CE%BC%CF%80%CE%BF%CF%82) is fields in the sense of the flatland countryside in general, not specific plots of land like a χωράφι.