Well, if you don't subtract the one to get to the Mersenne primes, the same logic would suggest that there are infinitely many primes of the form $2^n$, no?

I haven't given this too much thought, but I would've thought the natural approach would've been to write the conjugates in terms of the standard power basis, and then show that the matrix of coefficients was invertible.

Have you tried writing down some of its conjugates?

Better late than never, by several orders of magnitude. Thanks again.

This is excellent, and exactly the kind of answer I was looking for. Your condition in the lemma is eminently checkable. Now to remember why I wanted to know this 5 years ago...