# David Hilbert's paper: Substitution of the group of cyclotomic field

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?

• 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. Commented Jan 26 at 15:48
• @TorstenSchoeneberg, maybe make your apt comment into an answer, so this question will be "resolved"... :) Commented Jan 29 at 21:16

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