5 Kummer’s Lemma
In this section we prove Theorem 4.8. The proof follows [ SD01 ] . We begin with some lemmas that will be needed in the proof.
Let \(K\) be a number fields and \(\alpha \in K\). Then there exists a \(n \in \mathbb {Z}\backslash \{ 0\} \) such that \(n \alpha \) is an algebraic integer.
See mathlib.
Let \(K/F\) be a Galois extension of number fields whose Galois group \(\operatorname{Gal}(K/F)\) is cyclic with generator \(\sigma \). If \(\alpha \in K\) is such that \(N_{K/F}(\alpha )=1\), then
for some \(\beta \in \mathcal{O}_K\).
Pick some \(\gamma \in K\) and consider
where \([K/F]=n\). Since the norm \(\alpha \) is \(1\) then we have \(\alpha \sigma (\alpha )\cdots \sigma ^{n-1}(\alpha )=1\) from which one verifies that \(\alpha \sigma (\beta )=\beta \). This also uses that since we know that Galois group is cyclic, all the embeddings are given by \(\sigma ^i\). Note that \( \alpha + \sigma \alpha \sigma (\alpha )+\cdots +\sigma ^{n-1}\alpha \sigma (\alpha )\cdots \sigma ^{n-1}(\alpha )\) is a linear combination of the embeddings \(\sigma ^i\). Now, using 2.4 we can check that they must be linearly independent. Therefore, it is possible to find a \(\gamma \) such that \(\beta \neq 0\). Putting everything together we have found \(\alpha =\beta /\sigma (\beta )\). Using 5.1 we see that its possible to make sure \(\beta \in \mathcal{O}_K\) as required.
Let \(K/F\) be a Galois extension of \(F=\mathbb {Q}(\zeta _p)\) with Galois group \(\operatorname{Gal}(K/F)\) cyclic with generator \(\sigma \). Then there exists a unit \(\eta \in \mathcal{O}_K\) such that \(N_{K/F}(\eta )=1\) but does not have the form \(\epsilon /\sigma (\epsilon )\) for any unit \(\epsilon \in \mathcal{O}_K\).
This is Hilbert theorem 92 ( [ HLA\(^{+}\)98 ] ), also Lemma 33 of Swinnerton-Dyer’s book ( [ SD01 ] ).