One good book established that woman (6)
I believe the answer is:
esther
'one good book' is the definition.
(I've seen this before)
'established that woman' is the wordplay.
'established' becomes 'est' (abbreviation).
'that woman' becomes 'her' ('her' refers to a woman).
'est'+'her'='ESTHER'
(Other definitions for esther that I've seen before include "There's a girl from the Old Testament" , "Biblical heroine, who married the king of Persia" , "Biblical heroine who became queen of Persia" , "Section of Bible" , "Holy Book" .)