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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An octic reciprocity law of Scholz type

Authors: Duncan A. Buell and Kenneth S. Williams
Journal: Proc. Amer. Math. Soc. 77 (1979), 315-318
MSC: Primary 10A15; Secondary 12A45
MathSciNet review: 545588
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Abstract: The authors [3] have conjectured that if p and q are distinct primes satisfying

$\displaystyle p \equiv q \equiv 1\quad \pmod 8,\quad {(p/q)_4} = {(q/p)_4} = + 1,$


$\displaystyle {\left( {\frac{p}{q}} \right)_8}{\left( {\frac{q}{p}} \right)_8} ... ...uad {\text{if}}\;N({\varepsilon _{pq}}) = + 1,} \hfill \\ \end{array} } \right.$

where $ {\varepsilon _p}$ is the fundamental unit of $ Q(\sqrt p ),N({\varepsilon _{pq}})$ denotes the norm of the unit $ {\varepsilon _{pq}}$, and $ h(pq)$ is the class number of $ Q(\sqrt {pq} )$. A proof of this conjecture is given, which makes use of results of Bucher [2].

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Keywords: Legendre symbol, rational reciprocity laws, Scholz's law, fundamental unit of real quadratic field, class numbers of quadratic fields
Article copyright: © Copyright 1979 American Mathematical Society

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