Dissolute prince goes into business with American astronomer (10)
I believe the answer is:
copernicus
'astronomer' is the definition.
(I've seen this before)
'dissolute prince goes into business with american' is the wordplay.
'dissolute' indicates anagramming the letters.
'goes into' indicates putting letters inside.
'business' becomes 'co' (short for company).
'with' says to put letters next to each other.
'american' becomes 'US' (abbreviation for United States).
'prince' is an anagram of 'pernic'.
'co'+'us'='cous'
'pernic' inserted into 'cous' is 'COPERNICUS'.
(Other definitions for copernicus that I've seen before include "Polish astronomer d. 1543" , "Famous Prussian" , "16th century astronomer" , "Polish scholar" , "He said we move in a ring" .)