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Why is Kronecker's Jugendtraum only for abelian extensions?

Why is Kronecker's Jugendtraum only for abelian and not for more general extensions of number fields? Wikipedia, Hilbert's Twelfth Problem.
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The Class Number One Problem for Real Quadratic Fields

An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick ...