I'm writing a fanfic story that is based off Stargate SG-1 and I cannot find this information.
How can a Goa'uld be extracted from its host and who can do it?
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It has never been explained how, but the removal process is exceedingly difficult. The symbiote can and will kill the host if it is subject to removal and a dead symbiote (unless it takes extraordinary steps to prevent it) kills the host automatically.
The Tok'Ra and Tollan showed evidence of technology which could remove the symbiote safely (though the Tok'Ra's process was not perfect, it had a significant element of risk). It can be assumed that the Asgard could do it as well, given their medical expertise. It is safe to assume that ascended beings could remove symbiotes, but it is not known if Priors can do so.
As of the end of SG-1, no other species had demonstrated this ability, and it had never been done on Earth.