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The sum of algebraic elements over a field is algebraic. Is there a way to write down an explicit equation of algebraic dependence for it, knowing the equations of algebraic dependence for the individual elements?

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Have you tried writing down an equation for $\sqrt{2} + \sqrt[3]{3}$ (or even, $\sqrt{2} + \sqrt{3}$)? –  Srivatsan Dec 4 '11 at 18:19
    
@Srivatsan: It doesn't seem difficult on a case by case basis. In your examples, I can isolate one of the radicals and keep raising to the appropriate powers. For example, with $x=\sqrt{2}+\sqrt{3}$, we have $(x-\sqrt{2})^2=3$. So $(x^2+1)^2=(2\sqrt{2}x)^2$ –  Tomoki Visawa Dec 4 '11 at 18:26
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Use tensor products. See Theorem 2.3 and Example 2.4 at math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod2.pdf. –  KCd Dec 4 '11 at 18:38
    
@KCd: Thank you. This is perfect. –  Tomoki Visawa Dec 4 '11 at 18:42

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