It should be either "X's theorem" or "the X theorem," not just "X theorem". There is no single rule saying just one of those is always correct. Sometimes both are and sometimes just one is.
Eisenstein's criterion or the Eisenstein criterion, Mordell's conjecture or the Mordell conjecture, Chebotarev's density theorem or the Chebotarev density theorem are all standard English. You can go either way.
What qualifies as correct English is what people who speak (native) English deem to be correct English. There is no way to apply logic to the matter. The answer to your question is that you should write whatever the common convention is, which you find out by reading and talking to people in the field. It could go both ways (see above) or only one way is standard and you should just use that. For example, it is standard to refer to "the Weil conjectures". Nobody these days talks or writes about "Weil's conjectures" for the set of conjecture he made about varieties over finite fields.