Ideally when there is such breakage of a tooth, and it involves the exposure of the tooth nerve: we advise a root canal treatment followed by a full coverage crown.
Which removes pain and restores function.
Being your wisdom tooth (if that is, the last tooth) it does not play much role in most patients being a vestigial tooth and hence are suggested in for extraction. Also if they are impacted (not fully erupted or tilted) extraction is the right choice. Ofcourse the root patterns of the concerned tooth and its proximity to the lower jaw nerve has to be examined for a careful uneventful extraction: however wisdom teeth post extraction need no replacement.
Also, incase the opposing wisdom tooth is present and they both form a bite: you could consider RCT and crown as an option: provided your mouth opening is adequate for the concerned instruments to reach the said tooth.
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Finally: If your dentist advises against a root canal treatment: either due to the tooth being impacted or having no opposing tooth or due to lack of space for the RCT instruments for an effective treatment: please go ahead and have the tooth extracted. If not: your other option is an RCT and crown.