Is it possible to write a detective investigation fiction novel which

Just as well as you can set a detective-investigation novel in 3000 AD, you can set it in 1000 BC. Unlike a story set in the future, you have a time that once existed, cultures that thrived...anybody who writes a story set in the past will have (if they cared anything about accuracy) researched the people and society in question.
 
Of course it is. It's been done, too. Loads of times. Even more if you go forward a few hundred years to ancient Rome.

http://en.wikipedia.org/wiki/Margaret_Doody#The_Aristotle_series
 
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