Ramifies as well as split

Question

Can somebody give an example of a finite Galois extension of $\mathbb{Q}$ where a rational prime $p$ ramifies ( some (equivalently every) prime lying over $p$ has ramification index $> 1$) as well as splits ( at least $2$ distinct primes lie over it)?

Answer 1

As suggested in the comment, we only need to construct one extension where $p$ ramifies and another extension where $p$ splits.

For example, $3$ ramifies in $\Bbb Q(\sqrt{3})$ and splits in $\Bbb Q(\sqrt{7})$, so we know that $e_3 = g_3 = 2$ in $K = \Bbb Q(\sqrt{3}, \sqrt{7})$.

According to PARI/GP, $\mathcal O_K = \Bbb Z[\alpha]$ where $\alpha = \sqrt{\dfrac{5 + \sqrt{21}}{2}} = \dfrac{\sqrt{3} + \sqrt{7}}{2}$, and $(3) = (3, 1+\alpha)^2 (3, 1-\alpha)^2$.

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Appendix: PARI/GP code

(02:25) gp > f = x^4 - 5*x^2 + 1;
(02:25) gp > nfbasis(f)
%2 = [1, x, x^2, x^3]
(02:25) gp > idealfactor(nfinit(f),3)
%3 =
[[3, [-1, 1, 0, 0]~, 2, 1, [-1, 2, 1, 5; 1, 1, -1, 1; 1, 1, -4, -2; 1, 1, -1, -2]] 2]

[ [3, [1, 1, 0, 0]~, 2, 1, [1, 2, 1, -5; 1, -1, 1, 1; 1, -1, 4, -2; -1, 1, -1, 2]] 2]