Sylow-conjugate number fields (Q6584666)

Let $K\subseteq \mathbb Q$ be a number field, i.e., a finite extension of the field of rational numbers $\mathbb Q$ embeded in a fixed algebraic closure $C$ of $\mathbb Q$. \textit{J. Neukirch} [J. Reine Angew. Math. 238, 135--147 (1969; Zbl 0201.05901)], \textit{M. Ikeda} [Arch. Math. 26, 250--252 (1975; Zbl 0316.12003)], \textit{K. Iwasawa} [Bull. Am. Math. Soc. 65, 183--226 (1959; Zbl 0089.02402)] and \textit{K. Uchida} [Tôhoku Math. J. (2) 31, 359--362 (1979; Zbl 0422.12006)] proved that a number field $K$ is determined by the structure of its absolute Galois group $\mathrm{Gal}(K)=\mathrm{Gal}(C/K)$. \NNamely, if $L$ is a number field with $\mathrm{Gal}(L)$ isomorphic to $\mathrm{Gal}(K)$ as profinite groups, then $L$ is isomorphic to $K$. Answering a question raised by \textit{F. Pop} [Lond. Math. Soc. Lect. Note Ser. 242, 113--126 (1997; Zbl 0917.14011)], the authors of this interesting article show that $K$ is not determined by the structure of the Sylow subgroups of $\mathrm{Gal}(K)$. In theire previous article the authors showed already that the structure of the maximal prosolvable quotient of $\mathrm{Gal}(K)$ determines $K$. Recent works of other researchers show that even much smaller quotients of $\mathrm{Gal}(K)$ determine $K$. Florian Pop [loc. cit.] raised the natural question whether the structure of the $p$-Sylow subgroups $\mathrm{Gal}(K)_{p}$ of $\mathrm{Gal}(K)$, where $p$ runs over all primes, already suffices to determine $K$? The article under review shows that this is not the case. The authors introduce the following concept. Two number fields $K$ and $L$ are said to be Sylow-conjugate if for every prime $p$, the $p$-Sylow subgroups of $\mathrm{Gal}(K)$ and $\mathrm{Gal}(L)$ are conjugate in $\mathrm{Gal}(\mathbb Q)$. In particular, the absolute Galois groups of two Sylow-conjugate number fields have isomorphic $p$-Sylow subgroups. Furthermore, two Sylow-conjugate number fields $K,L$ have the same degrees $[K:\mathbb Q]=[L:\mathbb Q]$ and the same Galois closure $M$ over $\mathbb Q$. The authors give many examples for which $K$ and $L$ are not isomorphic to each other. Thus, for example, the following pairs $(K,L)$ are Sylow-conjugate but not isomorphic: \N\begin{enumerate}\N\item $K=\mathbb Q(\alpha)$ and $L=\mathbb Q(\beta)$, where $\alpha$ and $\beta$, respectively, are the roots of: $p_7(x)=x^7-7x+3$ and $q_7(x)=x^7+14x^4-42x^2-21x+9$. \N\item $K=\mathbb Q(\alpha)$ and $L=\mathbb Q(\beta)$, where $\alpha$ and $\beta$, respectively, are the roots of: $p_8(x)=x^8-4x^7-4x^6+26x^5+2x^4-52x^3+31x+1$ and $q_8(x)=x^8+12x^7+30x^6-108x^5-402x^4+342x^3+1256x_2-687x-337$.\end{enumerate}