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So I want to find examples of $K/L$ field extensions with $G(K/L)=\mathbb{Z}/n$ and $A_n$.

For the first case I just take $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$. I am just looking for an example for the $A_n$ case, I will prove that indeed the provided example works.

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A generic example is $K(t_1,\ldots,t_n) / K(s_1, \ldots, s_n, \sqrt{\Delta})$ where $K$ is a field with $\text{char}(K) \neq 2$; $\{t_1, \ldots, t_n\}$ is an algebraically independent set over $K$; $s_1, \ldots, s_n$ are the elementary symmetric polynomials in the indeterminates $t_1, \ldots, t_n$ (ie, such that $f = (X-t_1) \ldots (X-t_n) = X^n + \sum_{j=1}^{n-1} (-1)^j s_j X^{n-j}$); and $\displaystyle \Delta = \prod_{1 \leq i < j \leq n} (t_i - t_j)^2$. ($\Delta$ is called the discrimiminant of the polynomial $f$.)

For concrete examples where $L = \mathbb{Q}$ and $K$ is the splitting field of a polynomial $f \in \mathbb{Z}[X]$ of degree $n$, see the examples here.

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This is explained shortly in wikipedia for the inverse Galois problem. A book detailing such topics as Hilbert's Irreducibility Theorem is Serre's "Topics in Galois Theory", freely available at Darmon's webpage.

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