## 5.4 Some Ramification results

We will need several results about ramification in degree \(p\) extensions of \(\mathbb {Q}(\zeta _p)\). Following [ SD01 ] we do this by using the relative different ideal of the extension.

Let \(K, F\) be number fields with \(F \subseteq K\). Let \(A\) be an additive subgroup of \(K\). Let

and

The relative different \(\mathfrak {d}_{K/F}\) of \(K/F\) is then defined as \(((\mathcal{O}_K)^*)^{-1}\) which one checks is an integral ideal in \(\mathcal{O}_K\). This is also the annihilator of \(\Omega ^1_{\mathcal{O}_K/\mathcal{O}_F}\) if we want to be fancy.

Let \(K/F\) be an extension of number fields and let \(S\) denote the set of \(\alpha \in \mathcal{O}_K\) be such that \(K=F(\alpha )\). Then

where \(m_\alpha \) is denotes the minimal polynomial of \(\alpha \).

This is [ SD01 , Theorem 20 ]

Let \(K/F\) be an extension of number fields with \(\mathfrak {p}_F, \mathfrak {p}_K\) prime ideas in \(K,F\) respectively, with \(\mathfrak {p}_K^e \parallel \mathfrak {p}_F\) for \(e {\gt}0\). Then \(\mathfrak {p}_K^{e-1} \mid \mathfrak {d}_{K/F}\) and \(\mathfrak {p}_K^{e} \mid \mathfrak {d}_{K/F}\) iff \(\mathfrak {p}_F \mid e\).

[ SD01 , Theorem 21 ]

Let \(K\) be a number field, \(p\) be a rational prime below \(\mathfrak {p}\) and let \(\alpha _0, \xi \) be in \(\mathcal{O}_\mathfrak {p}^\times \) (the units of the ring of integers of the completion of \(K\) at \(\mathfrak {p}\)) be such that

where \(\mathfrak {p}^m \parallel p\) and \(r(p-1){\gt}m\). Then there exists an \(\alpha \in \mathcal{O}_{\mathfrak {p}}^\times \) such that \(\alpha ^p =\xi \).

This is [ SD01 , Lemma 20 ]

Let \(F=\mathbb {Q}(\zeta _p)\) with \(\zeta _p\) a primitive \(p\)-th root of unity. If \(\xi \in \mathcal{O}_F\) and coprime to \(\lambda _p:=1-\zeta _p\) and if

for some \(\alpha _0 \in \mathcal{O}_{F_\mathfrak {p}}^\times \) (the ring of integers of the completion of \(F\) at \(\lambda _p\)), then the ideal \((\lambda _p)\) is unramified in \(K/F\) where \(K=F(\sqrt[p]{\xi })\).

Suppose that we have \(\xi = \alpha _0^p \mod \lambda _p^{p+1}\). The using 5.8 with \(m=p-1,r=2\) gives an \(\alpha \) such that \(\alpha ^p=\xi \) in \(\mathcal{O}_{F_\mathfrak {p}}^\times \). Meaning that \((\lambda _p)\) is split in the local extension and hence split in \(K/F\) (note: add this lemma explicitly somewhere).

If we instead have \(\lambda _p^p \parallel (\xi -\alpha _0^p)\), then pick some element in \( K\) which agrees with \(\alpha _0\) up to \(\lambda _p^{p+1}\), which we again call \(\alpha _0\) and consider

so that \(K=F(\eta )\). Now, if \(m_\eta \) is the minimal monic polynomial for \(\eta \) then by definition it follows that

So in fact \(\eta \in \mathcal{O}_K\). Now, if we look at \(m_\eta '\) we see that \(m_\eta '(\eta )\) is coprime to \(\lambda _p\) and therefore coprime to \(\mathfrak {d}_{K/F}\) (by 5.6) and therefore \(\lambda _p\) is unramified (by 5.7).