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 \(p|a_3(p)\).


11R37 Class field theory
11R23 Iwasawa theory
11R20 Other abelian and metabelian extensions
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