# zbMATH — the first resource for mathematics

Ramified characters of idèle groups of one-class quadratic fields. (English. Russian original) Zbl 0956.11027
Proc. Steklov Inst. Math. 224, 107-114 (1999); translation from Tr. Mat. Inst. Steklova. 224, 122-129 (1999).
The author considers ramified characters $$\theta$$ of the idele group $$A_d^\times$$ of a one-class quadratic field $$\mathbb Q(\sqrt{d})$$. Necessary and sufficient conditions are found for a character $$\theta$$ to be trivial on the multiplicative group $$\mathbb Q^\times (\sqrt{d})$$. Thus a complete description of characters of the group $$A_d^\times /\mathbb Q^\times (\sqrt{d})$$ is found.
Note that in a subsequent paper [V. S. Vladimirov, Tr. Mat. Inst. Steklova 228, 76-89 (2000)] the author used this result in order to obtain adelic formulas for gamma and beta functions of one-class quadratic fields.
For the entire collection see [Zbl 0942.00074].

##### MSC:
 11R56 Adèle rings and groups 11R11 Quadratic extensions 22B99 Locally compact abelian groups (LCA groups)
##### Keywords:
idele group; quadratic field; multiplicative character