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Given a finite Galois extension of number fields $L/K$ with Galois group $G$ and a surjection $E\twoheadrightarrow G$ of finite groups, the Galois embedding problem is the question of whether there exists a Galois extension $M/K$ containing $L$ such that $Gal(M/K)\cong E$.

Let's assume we can solve our Galois embedding problem. Are there common applications to such a result?

Do you know of an example where the solution of such a problem implied an interesting/seemingly unrelated result?

Thank you.

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Shafarevich made heavy use of embedding problems in his resolution of the inverse Galois problem for solvable groups. The (naive) viewpoint is that solving the relevant embedding problems allowed him to reduce to the case of abelian extensions, where a positive solution to the inverse Galois problem was already known.

You can find further details and precise statements in Sections IX.4 and IX.5 of:

Neukirch, Schmidt, Wingberg - Cohomology of number fields.

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