What ideas about democracy did the United States incorporate into government?
The United States incorporated voting into government. They also incorporated who should vote like no women, children only men could vote and so on and so forth.
What were the field of studies explored by the Greeks and the significant accomplishments under these studies?
Some of the fields studied by the Greeks are Science and Psychology. Some of the most important discoveries was about the Moon the Sun and the Earth. One really famous discoveries was that the earth orbits the sun not the other way around.