Show that 2 has ramification index 4 on field

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I'm stuck at the item (b) in this quadratic number field question:

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I tried writing $\mathfrak{P}$ as $(1+i) \mathcal{O}_K + (1+\alpha) \mathcal{O}_K$ (that is the ideal generated by two elements, right?) and squaring it, but couldn't find anything that would solve this problem.