Geodesic
Ramified Characters of Id\`ele Groups of One-Class Quadratic Fields
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra. Topology. Differential equations and their applications, Tome 224 (1999), pp. 122-129

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The necessary and sufficient conditions for the triviality of the ramified characters of the idèle group $A_d^\times $ of a one-class quadratic field $\mathbb Q(\sqrt d)$ on its multiplicative group $\mathbb Q^\times(\sqrt d)$ are established; i.e., the characters of the group $A_d^\times/\mathbb Q^\times(\sqrt d)$ are completely described. 
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     author = {V. S. Vladimirov},
     title = {Ramified {Characters} of {Id\`ele} {Groups} of {One-Class} {Quadratic} {Fields}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {122--129},
     publisher = {mathdoc},
     volume = {224},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_1999_224_a6/}
}
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V. S. Vladimirov. Ramified Characters of Id\`ele Groups of One-Class Quadratic Fields. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra. Topology. Differential equations and their applications, Tome 224 (1999), pp. 122-129. http://geodesic.mathdoc.fr/item/TM_1999_224_a6/