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Class Numbers and Biquadratic Reciprocity

Published online by Cambridge University Press:  20 November 2018

Kenneth S. Williams  and
James D. Currie
Kenneth S. Williams
Affiliation:
Carleton University, Ottawa, Ontario
James D. Currie
Affiliation:
Carleton University, Ottawa, Ontario
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0. Notation. Throughout this paper p denotes a prime congruent to 1 modulo 4. It is well known that such primes are expressible in an essentially unique manner as the sum of the squares of two integers, that is,

(0.1)

with |a| and |b| uniquely determined by (0.1). Since a is odd, replacing a by –a if necessary, we can specify a uniquely by

(0.2)

Further, as {[(p – l)/2]!}2 = – 1 (mod p), we can specify b uniquely by

(0.3)

These choices are assumed throughout.

The following notation is also used throughout the paper: h(d) denotes the class number of the quadratic field of discriminant d, (d/n) is the Kronecker symbol of modulus |d|, [x] denotes the greatest integer less than or equal to the real number x, and {x} = x – [x].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 4 , 01 August 1982 , pp. 969 - 988
Copyright
Copyright © Canadian Mathematical Society 1982

References

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