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June 15

Monster group

The article Monster group mentions "The monster can be realized as a Galois group over the rational numbers".

Is that a deep or interesting result? There is a citation that I didn't try to read. Would it follow immediately from the inverse Galois problem if that were solved in the affirmative?
It means that there is a specific minimal rational polynomial whose splitting field's automorphism group is exactly the Monster, amirite? Is there a reasonable way to write the polynomial down? What would its degree be? Would it be somehow interesting?
What is the current general status of the inverse Galois problem, other than "open"? The article doesn't say much. By that I mean is it like some other open problems that have seen steady progress and might have an end in sight, or is it just a mystery? I remember being surprised in math class when the teacher said the problem was open. It sounded like something that should be simple, like maybe an exercise we'd get later in the class. Nope.

Update: I just found a MO thread from some years back, about the Monster and the IGP, and in particular an answer sketching how the Monster was proved to be a Galois group. 2601:648:8202:350:0:0:0:90B2 (talk) 03:40, 15 June 2022 (UTC)[reply]