What do we know about the American Dream as understood in The Great Gatsby, and why is it important?

The "American Dream" as illustrated in The Great
Gatsby is the idea that many American's extrapolate from the part of the
Constitution which guarantees all citizens the rights of "...life, liberty, and the
pursuit of happiness." The document only promises that Amerians will have the right to
go after happiness. It doesn't promise that happiness will
come.

However, many are disappointed when they don't
achieve this happiness, because they operated under the premise that America was the
land of opportunity (which they interpreted as a promise of the prosperity and
success.)

It is this dream (that every American would have
a loving family, success, and material wealth) that is explored in the Great
Gatsby. Gatsby spends his life chasing after wealth so that he can get his
love and his happy home. So, in a sense, he comes to symbolize all those who place their
happiness on having these things.

When Gatsby is killed at
the end of the book, I believe it is Fitzgerald's commentary that the American Dream
also died. I think he meant to tell us that chasing after wealth and a finite definition
of happiness doesn't bring happiness at all...only disappointment and
heartache.