Note on the Units of a Real Quadratic Field
نویسنده
چکیده
(here b and a run over the quadratic nonresidues and quadratic residues, respectively, that lie between 0 and p) is a unit of the real quadratic field R(y/p), and that 77> 1. The fact that -n > 1 is usually deduced from the theory of the classnumber of quadratic fields. We present a short proof independent of the theory of the class-number. As in the paper of Chowla and Mordell [Note on the nonvanishing of L(l), Proc. Amer. Math. Soc. 12 (1961), 283-284], we have tj^I since 7,(1)^0 (proved in the paper cited) in the relation [ibid. ] (1) r, = exp{S7(l)}, where P(l) = X¡i" (n|/>)»-1 and S = Zï (« | ¿>)0". Here (n\p) is Legendre's symbol. From the theory of the Gaussian sum,
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