Rationalizing denominator with any number of radicals

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

• i think this form is much simpler that its rationalized form if it exits!!!!! – logo May 21 at 21:22
• 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. – dantopa May 21 at 21:22
• chech this question math.stackexchange.com/questions/2240818/… – logo May 21 at 21:26
• 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. – user10354138 May 21 at 21:34
• Symmetric function theorem. – user10354138 May 21 at 21:43