---

Statement

Satz (§ 6, p. 225). Let $K$ be a normal field (written in Fraktur in the print) that contains the $l$-th roots of unity, and let $\alpha_1,\ldots,\alpha_t$ be integers of $K$ such that the product

$$\alpha_1^{m_1}\alpha_2^{m_2}\cdots\alpha_t^{m_t}\qquad(100)$$

can be the $l$-th power of a number of $K$ only when each of $m_1,\ldots,m_t$ is divisible by $l$. Let $\zeta=e^{2\pi i/l}$ and let $c_1,\ldots,c_t$ be arbitrarily prescribed numbers from $0,1,\ldots,l-1$. Then $K$ contains infinitely many prime ideals $\mathfrak P$ with

$$\left\{\frac{\alpha_1}{\mathfrak P}\right\}=\zeta^{c_1},\quad \left\{\frac{\alpha_2}{\mathfrak P}\right\}=\zeta^{c_2},\quad\ldots,\quad \left\{\frac{\alpha_t}{\mathfrak P}\right\}=\zeta^{c_t}.\qquad(101)$$

Here $\left\{\frac{\alpha}{\mathfrak P}\right\}$ is the generalized residue symbol: it equals $\zeta^c$ when $\alpha^{(p^f-1)/l}\equiv\zeta^c\pmod{\mathfrak P}$, where $f$ is the order (degree) of $\mathfrak P$. The paper remarks that $l$ divides $p^f-1$ because $K$ contains the field of $l$-th roots of unity.

The paper introduces the result as a theorem that Hilbert proved in a less sharp form (Zahlbericht, p. 426, Satz 152). The statement does not say that $l$ is prime; the proof uses it (p. 226: "da $l$ Primzahl ist").

Proof pointer

§ 6, pp. 226--228. Only prime ideals of degree $f=1$ are used. With $\beta_i=\sqrt[l]{\alpha_i}$ and $K$ taken as the field of rationality, the group of $K(\beta_1,\ldots,\beta_t)$ is shown to have order $l^t$ (pp. 226--227): if it were smaller, one of the equations (102), $z^l-\alpha_\nu=0$, would factor over $K(\beta_1,\ldots,\beta_{\nu-1})$, and this forces a relation (105), $\alpha_\nu=A^l\alpha_1^{c_1}\cdots\alpha_{\nu-1}^{c_{\nu-1}}$ with $A$ in $K$, excluded by the hypothesis (100). The group consists of the substitutions $S_1^{\xi_1}\cdots S_t^{\xi_t}$, where $S_i$ multiplies $\beta_i$ by $\zeta$ and fixes the other $\beta_j$. These substitutions also occur in the group of the normal closure (the "Norm") of $K(\beta_1,\ldots,\beta_t)$ over the rationals, and the paper infers (p. 227) that infinitely many rational primes belong to the class of $S=S_1^{c_1}\cdots S_t^{c_t}$; the step cites no earlier result by number, and it is the class density of § 5 (equation (99)) that supplies it. Since $S$ fixes $K$, these primes split in $K$ into prime ideals of first degree; a prime ideal $\mathfrak P$ of $K$ lying under a prime ideal of the normal closure that belongs to $S$ satisfies (107) and hence (108), and the congruences (108) are equalities because distinct $l$-th roots of unity are incongruent modulo $\mathfrak P$ (p. 228). Not reconstructed here.

Read depth

Claims checked: the statement, the remark on $p^f-1$ and the proof on pp. 225--228 were read clause by clause on the page images of the print. Nothing here is independently reviewed.

Dependencies

Equation (99) of the same paper, applied over the rationals to the normal closure of $K(\beta_1,\ldots,\beta_t)$; the paper does not cite it by number.

Source. N. Tschebotareff, Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören, Math. Ann. 95 (1926), 191--228, doi:10.1007/BF01206606; the edition read is named on the source card.

Bears on

None: the paper mentions no Erdős problem, and no problem page cites it.