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I'm trying to develop a java class for exact algebraic numbers. I've come to a little bit of a roadblock as far as this goes. My question right now is how to rationalize these equations, but no-one I've asked seems to know.

Rationalize $$\frac{1}{a+\sqrt[c_1]{b_1}+\sqrt[c_2]{b_2}+\cdots+\sqrt[c_n]{b_n}}$$

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    $\begingroup$ i think this form is much simpler that its rationalized form if it exits!!!!! $\endgroup$ – logo May 21 at 21:22
  • $\begingroup$ Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax. $\endgroup$ – dantopa May 21 at 21:22
  • $\begingroup$ chech this question math.stackexchange.com/questions/2240818/… $\endgroup$ – logo May 21 at 21:26
  • $\begingroup$ Assuming $c_i\in\mathbb{N}$. In general, you have $$ \frac{1}{a+\sqrt[c_1]{b_1}+\sqrt[c_2]{b_2}+\dots+\sqrt[c_n]{b_n}} =\frac{\prod\limits_{(m_1,\dots,m_n)\neq 0}(a+\zeta_{c_1}^{m_1}\sqrt[c_1]{b_1}+\zeta_{c_2}^{m_2}\sqrt[c_2]{b_2}+\dots+\zeta_{c_n}^{m_n}\sqrt[c_n]{b_n})}{\prod\limits_{(m_1,\dots,m_n)\in C_{c_1}\times\dots\times C_{c_n}}(a+\zeta_{c_1}^{m_1}\sqrt[c_1]{b_1}+\zeta_{c_2}^{m_2}\sqrt[c_2]{b_2}+\dots+\zeta_{c_n}^{m_n}\sqrt[c_n]{b_n})}. $$the denominator is a polynomial in $a,b_1,\dots,b_n$ with integer coefficients. $\endgroup$ – user10354138 May 21 at 21:34
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    $\begingroup$ Symmetric function theorem. $\endgroup$ – user10354138 May 21 at 21:43

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