Hi sir....    First and foremost You have to take X_RAY of the particular tooth region to check the amount of Root remains..and the treatment will be based on the X_ray investigation only....if the root stump remains are only few millimeter it will resorb by itself otherwise  It is better to remove root stump of premolar to prevent furthur infection...        

It's better to get the tooth removed, otherwise you might develop infection later. And also, the neighbouring teeth might move into the space if you don't get it treated immediately..this might result in later difficulty in fixing new artificial tooth in that space

Without seeing x ray it will be difficult to say about whether that root piece will come normally or sugicslly. But in general very rare chances of premolar to come out via durgically

Hello, Although your tooth isnt hurting, still go to a dentist and get it extracted. It wont be a much of a tedious procedure. The dentist will anesthetize the area and you wont feel any pain thereafter.

If it is broken below the gum level and decayed then your dentist will give an incision to remove it. Surgical extraction is indicated in some cases. You can go ahead with it as unless you remove it , restoration is not possible