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I have a basic question in Galois theory. For any given natural number $n$ is there a Galois extension of $\mathbb Q$ of degree $n$?

I want to show that there are splitting fields of polynomials in $\mathbb Q[X]$ of arbitrary degrees over $\mathbb Q$.

A hint would be appreciated- many thanks.

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We can use basic inverse Galois theory to solve this problem. Let $p$ be a prime such that $p \equiv 1 \pmod n$, and let $\mu_p$ be the $p$th roots of unity. The cyclotomic extension $\mathbb{Q}(\mu_p)$ has Galois group $(\mathbb{Z}/p\mathbb{Z})^*$, which is cyclic with order $p - 1$. Since $n|p - 1$, there is a subgroup $H \subset \mathrm{Gal}(\mathbb{Q}(\mu_p)/\mathbb{Q})$ with order $(p - 1)/n$. By Galois theory, the field $K \subset \mathbb{Q}(\mu_p)$ fixed by $H$ has order $n$ and is Galois.

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  • $\begingroup$ I don't think there is a difference between using Galois theory and using inverse Galois theory. Does that make Galois theory an involution as an operator on mathematical problems? $\endgroup$ – RghtHndSd May 7 '15 at 16:55
  • $\begingroup$ I suppose I meant "theory associated with the inverse Galois problem". $\endgroup$ – Benjamin Peterson May 7 '15 at 16:55

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