As you probably know, the gum boil you mention is often a sign of infection associated with a tooth. In your case, one where bacteria live in the space between the two broken portions.
Bacteria can be removed from root canals because instruments can be worked down them.
In the case of a crack, there are no tools that can negotiate that type of space, thus the infection/bacteria can't be cleared up.
That is why your dentist has recommended tooth extraction.

Lower molars are two-rooted teeth.
If your dentist feels confident about which root is cracked (the position of the gum boil might identify this), it's conceivable that that root can be cut off (root amputation).

There are a lot of associated issues:
Does the dentist have adequate access to be able to perform the surgery/remove the root?
Are the molar's roots entirely separate or instead fused?
Considering the amount of chewing forces involved, with just one root, can the tooth adequately withstand them? (If not, it will loosen and/or shift over time.)
After the root amputation, will the architecture (contours) of the bone and gum tissue be such that the tooth and/or neighboring teeth can be cleansed properly (so to avoid gum disease or decay)?
If you have an existing crown, can it be retained or will a new one need to be made?

It's not all that often when all of these conditions can be met.
And while not rare, using this route is not all that common either.
But your only hope of making a go of a tooth with a cracked root will need to involve root amputation, so ask about that possibility.

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