Fermat’s Last Theorem for regular primes: Some Ramification results

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.

Definition 5.5 ✓

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

\[ A^{-1}=\{ \alpha \in K | \alpha A \in \mathcal{O}_K\} \]

and

\[ A^* = \{ \alpha \in K | \operatorname {Tr}_{K/F} (\alpha A) \in \mathcal{O}_F\} . \]

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.

Lemma 5.6 ✓

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

\[ \mathfrak {d}_{K/F} = \left(m_{\alpha }'(\alpha ) : \alpha \in S \right) \]

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

Proof ▼

This is [ SD01 , Theorem 20 ]

Lemma 5.7 ✓

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 \(\mathfrak {p}_F \mid e\).

Proof ▼

[ SD01 , Theorem 21 ]

Lemma 5.8

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

\[ \alpha _0^p \equiv \xi \mod \mathfrak {p}^{m+r} \]

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 \(\alpha ^p =\xi \).

Proof ▼

This is [ SD01 , Lemma 20 ]

Lemma 5.9 ✓

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

\[ \xi \equiv \alpha _0^p \mod \lambda _p^p \]

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 \(K=F(\sqrt[p]{\xi })\).

Proof ▼

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

\[ \eta = (\sqrt[p]{\xi }-\alpha _0)/\lambda _p \]

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

\[ m_\eta (x) \equiv x^p+\left(\alpha _0^{p-1}p/\lambda _p^{p-1} \right)x+ (\alpha _0^p-\xi )/\lambda _p^p \mod \lambda _p \]

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).