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I created inverse Galois problem. However as it stand this stub does not meet nlab standards since it doesn’t refer to category theory.
Is that a joke? (I mean the implication that an article must refer to category theory in order to “meet nlab standards”.)
To the best of my knowledge, this is unsolved and is considered a quite seriously hard problem.
If you think you have a proof that every finite group occurs as the Galois group of some Galois extension of the rationals, then you should submit it to the Annals of Math. or some other appropriate top-notch journal and get it published. If it gets accepted, then you will become very famous indeed.
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