In this paper, we find some properties of Euler’s function and Dedekind’s function. We also generalize these results, from an algebraic point of view, for extended Euler’s function and extended Dedekind’s function, in algebraic number fields. Additionally, some known inequalities involving Euler’s function and Dedekind’s function, we generalize them for extended Euler’s function and extended Dedekind’s function, working in a ring of integers of algebraic number fields.

中文翻译：

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扩展的Euler函数和扩展的Dedekind函数的一些性质

在本文中，我们发现了Euler函数和Dedekind函数的一些性质。我们还从代数的角度概括了在代数数域中扩展的欧拉函数和扩展的Dedekind函数的这些结果。此外，一些涉及Euler函数和Dedekind函数的已知不等式，我们将它们推广为扩展的Euler函数和扩展的Dedekind函数，在代数数域的整数环中工作。