A question about a notation in David Hilberts's "Ein neuer Beweis des Kroneckerschen Fundamentalsatzes über Abelsche Zahlkörper" (here a german online available source, not sure if there exist a free available english version.

Question: In Satz 1 Hilbert introduced the notion of $t:=(\Theta: \Theta^r)$ (where $\Theta:=e^{2 \pi i/l^h}$ a $l^h$-root of unity and $r$ certain natural number) and called it "Substitution der Gruppe des Kreiskörpers $k(\Theta)$", ie "substitution of the group of cyclotomic field $k(\Theta)$".

What's the meaning of $(\Theta: \Theta^r)$ in modern terms?

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    $\begingroup$ I think in more common notation of today, $t$ is just the element $\zeta \mapsto \zeta^r$ of the Galois group of the cyclotomic field. $\endgroup$ Jan 26 at 15:48
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    $\begingroup$ @TorstenSchoeneberg, maybe make your apt comment into an answer, so this question will be "resolved"... :) $\endgroup$ Jan 29 at 21:16

1 Answer 1


I think in modern notation, the element $\Theta$ (which is literally defined as $\displaystyle e^{\frac{2\pi i}{l^h}}$ i.e. a primitive $l^h$-th root of unity; $l$ is a prime here) would often be called $\zeta$ or $\zeta_{l^h}$, the field $k(\Theta)$ is really just the cyclotomic field extension $\mathbb Q(\zeta)$ of $k=\mathbb Q$, and for a given integer $r$ not divisible by $l$, $t$ (which one should maybe call $t_r$) is the element of its Galois group given by $\zeta \mapsto \zeta^r$.

In a bit fancier and possibly more precise modern terminology, the correspondence $r\mapsto [t_r: \zeta \mapsto \zeta^r]$ defines an isomorphism

$$(\mathbb Z/l^h \mathbb Z)^\times \rightarrow Gal(\mathbb Q(\zeta) \vert \mathbb Q).$$


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