2. For every group there exists a corresponding field extension (this is a consequence of the fundamental theorem of Galois theory).
Just a nit, but when talking about extensions of Q, this is called the Inverse Galois Problem and it is still an open problem.
That said, you don’t actually need this strong of a statement to show general insolvability of the quintic. Rather you just need to exhibit a single extension of Q with non-solvable Galois group. I believe adjoining the roots of something like x^5+x+2 suffices.
Just a nit, but when talking about extensions of Q, this is called the Inverse Galois Problem and it is still an open problem.
That said, you don’t actually need this strong of a statement to show general insolvability of the quintic. Rather you just need to exhibit a single extension of Q with non-solvable Galois group. I believe adjoining the roots of something like x^5+x+2 suffices.