Class numbers in the genus of quadratic forms, and algebraic groups

In this paper the class numbers $c(f)$ of quadratic forms $f$ with coefficients in an algebraic number field $K$ are studied by the methods of the theory of algebraic groups. It is shown that if a form $f$ is positive definite, then for any natural number $r$ there exists a quadratic form $g_r$, $K$-equivalent to $f$, such that $c(g_r)$ is divisible by $r$. A generalization of this result to semisimple algebraic $K$-groups of compact type is also obtained.
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