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I don't know how to solve the following equation: $x^5-h_1x^4-h_2x^3-h_5=0$, where $h_1=\beta_1+\beta_2$, $ h_2=\beta_1+\beta_2-\beta_1\beta_2-\frac{\beta_1\beta_2}{\beta_1+\beta_2}$, $h_5=\frac{(\beta_1\beta_2)^3}{(\beta_1+\beta_2)^2}$, where $1<\beta_1<\beta_2<\sqrt2$ , although I do know it is connected to the Galois theory....

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I've removed algebra tag, since we don't use algebra tag anymore, see meta for details. Perhaps you could use abstract-algebra or galois-theory, if you find these tags suitable for your question. –  Martin Sleziak Nov 20 '12 at 15:59

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