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Since every finite group $G$ is isomorphic to a subgroup of $S_{n}$ and according to the first answer on this question there is always (for all $n\geq 1$) a finite Galois extension $K/\mathbb{Q}$ with $\operatorname{Gal}(K/\mathbb{Q})\cong S_{n}$, doesn't the Fundamental Theorem of Galois Theory gives a positive answer to the Inverse Galois Problem?

Where is the obvious point I am missing? Thanks!

Shoutre
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    How would you go from an extension giving the symmetric group to one giving a specified subgroup? – Tobias Kildetoft Jul 30 '17 at 18:09
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    If $G\le S_n$ then, with $K$ as in your question, the Fundamental Theorem of Galois Theory implies that $$Gal(K/F)\simeq G,$$ where $F$ is the fixed field of $G$. Adjusting the situation to replace $F$ with $\Bbb{Q}$ is the challenge. – Jyrki Lahtonen Jul 30 '17 at 18:09

1 Answers1

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If $H\subset \mathrm{Gal}(K/\mathbb Q)$ then the set elements elements of $K$ fixed by $H$ is a field $k$, and you get $$H\cong\mathrm{Gal}(K/k)\,.$$

But the inverse Galois question is seeking $k$ so that $H\cong \mathrm{Gal}(k/\mathbb Q)$.

Mike Pierce
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Thomas Andrews
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    @Shoutre Don’t worry, it’s a fair question and a common misinterpretation (of either the main theorem of Galois theory or the inverse problem). – k.stm Jul 30 '17 at 18:27