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It is well known that the splitting fields of $x^3-2$ and $x^4-2$ over $\mathbf{Q}$ have Galois groups $D_6$ and $D_8$, Dihedral groups of $6$ and $8$ elements respectively. However, this pattern does not continue: the splitting field of $x^5-2$ over $\mathbf{Q}$ has a Galois group of order $20$ (in general for $p$ prime, the splitting field of $x^p-2$ over $\mathbf{Q}$ has Galois group $\mathbf{Z}_p\rtimes\mathbf{Z}_p^{\times}$).

My question is this: Is there any "reasonable" sequence $(p_n)$ of polynomials over $\mathbf{Q}$ such that the splitting fields of $p_n$ over $\mathbf{Q}$ have Galois groups $D_{2n}$?

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