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We prove that Hecke eigenvalues for any Hilbert and Siegel modular forms arealgebraic integers. Our method does not rely on cohomologicality nor Galoisrepresentations. We apply the integrality of Hecke eigenvalues for Hilbertmodular forms of non-parallel weight to the estimation of the growth of Heckefields of Hilbert cusp forms with non-vanishing central $L$-values. As afurther application, we give the growth of the fields of rationality ofcuspidal automorphic representations of ${\rm GL}_{2d}(\mathbb{A}_\mathbb{Q})$for a prime number $d$ with non-vanishing central $L$-values. We also apply theintegrality of Hecke eigenvalues for holomorphic Siegel cusp forms of generaldegree in order to give the growth of the Hecke fields of those forms.

Integrality of Hecke eigenvalues and the growth of Hecke fields