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CONGRUENCE RELATIONS FOR THE FUNDAMENTAL UNIT WITH NEGATIVE NORM OF A REAL QUADRATIC FIELD
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In this note, we give some explicit congruence relations for the fundamental unit with negative norm of a real quadratic field, by elementary method. Our results extensively solve the remaining case of the main theorem of Zhang and Yue, which only considers the case that real quadratic field has odd ideal class number and its fundamental unit has positive norm.
Abstract
In this note, we give some explicit congruence relations for the fundamental unit with negative norm of a real quadratic field, by elementary method. Our results extensively solve the remaining case of the main theorem of Zhang and Yue, which only considers the case that real quadratic field has odd ideal class number and its fundamental unit has positive norm.
참고문헌 (6)
- D. Chakraborty, "Congruence relations for the fundamental unit of a pure cubic eld and its class number", J. Number Theory, 166, pp.76~84 (2016)
- P. E. Conner, "Class Number Parity, World Sci. Ser. Pure Math., 8" (1988), World Scientific Singapore
- K. Ireland, "A Classical Introduction to Modern Number Theory, GTM" (1990), Springer-Velag
- F. Lemmermeyer, "Reciprocity Laws: From Euler to Eisenstein, Springer Monographs in Mathematics" (2000), Springer-Velag
- K. S. Williams, "Historical remark on a theorem of Zhang and Yue", J. Number Theory, 147, pp.691~693 (2015)
- Zhe Zhang, "Fundamental units of real quadratic fields of odd class number", J. Number Theory, 137, pp.122~129 (2014)
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