×

A note on the \(\mathbb Z_p\times\mathbb Z_q\)-extension over \(\mathbb Q\). (English) Zbl 1005.11060

For a nonempty set \(S\) of prime numbers, let \(Q^S\) denote an Abelian extension of \(Q\) whose Galois group over \(Q\) is topologically isomorphic to the direct product of the additive group of \(\ell\)-adic integers for all \(\ell\in S\). For a prime \(p\) let \(H_p(K)\) denote the Hilbert \(p\)-class field of the finite algebraic number field \(K\). Then \(H_p(Q^S)=\bigcup_k H_p(k)\) where \(k\) ranges over all finite algebraic number fields contained in \(Q^S\). The author notes that \(H_p(Q^S)= Q^S\) when \(S=\{p\}\). Therefore, the author considers the case \(S= \{p,q\}\). Let \(L_q\) denote the unique subfield of \(Q^s\) of degree \(q\), \(R_{p,q}\) denote the \(p\)-adic regulator of \(L_q\) and \(|\;|_p\) denote a normalized \(p\)-adic valuation. Then the author defines \(a_q(p)= p^{1-q} [H_p (L_q): L_q]|R_{p,q}|_p^{-1}\). When \(\{2,p\} \subset S\) and \(p^2\equiv 1\pmod{16}\) the author shows that \(H_p(Q^S)\) contains a subfield of degree \(a_2(p)\) over \(Q^S\). Similarly, if \(p\) and \(q\) are odd primes with \(p^{q-1}\equiv 1\pmod {q^2}\) and \(\{p,q\}\subset S\) then \(H_p(Q^S)\) contains a subfield of degree \(a_q(p)\) over \(Q^S\).
To complete his result the author shows that \(a_2(31)= 31\) and \(a_3(73)= 73\). Hence, if \(\{2,3,31,731\} \subset S\) then \(Q^S\) has an Abelian unramified extension of degree \(2263= 31\bullet 73\). The author also observes that 31 is the only prime \(p<20000\) satisfying \(p^2\equiv 1\pmod{16}\) and \(p|a_2(p)\). Similarly, 73 is the only prime \(p<10000\) satisfying \(p^2\equiv 1\pmod 9\) and \(p|a_3(p)\).

MSC:

11R37 Class field theory
11R23 Iwasawa theory
11R20 Other abelian and metabelian extensions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brumer, A.: On the units of algebraic number fields. Mathematika, 14 , 121-124 (1967). · Zbl 0171.01105 · doi:10.1112/S0025579300003703
[2] Fukuda, T., and Komatsu, K.: On the \(\lambda\) invariants of \(\mathbf{Z}_p\)-extensions of real quadratic fields. J. Number Theory, 23 , 238-242 (1986). · Zbl 0593.12003 · doi:10.1016/0022-314X(86)90093-4
[3] Hasse, H.: Über die Klassenzahl abelscher Zahlkörper. Academie, Berlin (1952); Springer, Berlin (1985). · Zbl 0046.26003
[4] Hatada, K.: Mod \(1\) distribution of Fermat and Fibonacci quotients and values of zeta functions at \(2-p\). Comment. Math. Univ. St. Pauli, 36 , 41-51 (1987). · Zbl 0641.12008
[5] Fröhlich, A.: On the absolute class-group of abelian fields. J. London Math. Soc., 29 , 211-217 (1954). · Zbl 0055.03302 · doi:10.1112/jlms/s1-29.2.211
[6] Iwasawa, K.: A note on class numbers of algebraic number fields. Abh. Math. Sem. Univ. Hamburg, 20 , 257-258 (1956). · Zbl 0074.03002
[7] Iwasawa, K.: On \(\Gamma\)-extensions of algebraic number fields. Bull. Amer. Math. Soc., 65 , 183-226 (1959). · Zbl 0089.02402 · doi:10.1090/S0002-9904-1959-10317-7
[8] Taya, H.: On \(p\)-adic zeta functions and \(\mathbf{Z}_p\)-extensions of certain totally real number fields. Tohoku Math. J., 51 , 21-33 (1999). · Zbl 0943.11049 · doi:10.2748/tmj/1178224850
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.