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?
Equation of algebraic dependence for sum of algebraic elements
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abstract-algebra
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1Have you tried writing down an equation for $\sqrt{2} + \sqrt[3]{3}$ (or even, $\sqrt{2} + \sqrt{3}$)? – 2011-12-04
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0@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 $(x^2+1)^2=(2\sqrt{2}x)^2$ – 2011-12-04
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1Use tensor products. See Theorem 2.3 and Example 2.4 at http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod2.pdf. – 2011-12-04
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0@KCd: Thank you. This is perfect. – 2011-12-04