How can we solve the following quintic in closed form ?
$$16x^5-200x^3-200x^2+25x+30=0$$
What is special about this equation? What can we say about solvability? I'm not particularly into math. I saw this equation in a discussion group. Here are some ideas about the equation from non-mathematicians .
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There is no closed-form formula for $5$-degree equations. Therefore there is no solution. In my opinion this is wrong. For example, $x^5=1$ can be easily solved. The true version if that idea would be there is no general solution formula . (by radicals)
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The equation is not factored. Therefore, there is no closed-form solution. This idea is also wrong in my opinion. Because the factors do not have to be rational numbers.
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Wolfram Alpha does not give closed form for the equation. Therefore, it cannot be solved. I cannot comment on this idea.
I found that the quintic is irreducible over $\Bbb Q$.
I've heard a little bit about Galois theory. But I don't have the mathematical knowledge. If possible, could you help to find out if this equation is solvable? The problem author argues that the equation has a solution.
I tried WA software several times. But, it does not succeed.
Best Answer
This polynomial $f$ has Galois group $F_5$, the Frobenius group of order 20. Since this group is solvable, then $f$ is solvable by radicals. Here are Magma commands showing this.
which give as output
The docstring for the command
SolveByRadicals
describes how it gives solutions to a solvable polynomial:(See here for more details.) We enter
which yields output
Denote by $\alpha_i$ the elements
K.i
for $i = 1, \ldots, 4$. Then this output is saying that we have a chain of simple radical extensions $$ K = K_1 \supseteq K_2 \supseteq K_3 \supseteq K_4 \supseteq \mathbb{Q} $$ where $K_i = K_{i+1}(\alpha_i)$ and the $\alpha_i$ satisfy the equations \begin{align*} 0&={\alpha_1}^5+1/512(-1875\alpha_4-3125)\alpha_3\alpha_2-2560000000\\ 0&={\alpha_2}^2+15118284881920000\alpha_4-34359738368000000\\ 0&={\alpha_3}^2+32\alpha_4+160\\ 0&={\alpha_4}^2-5\,.\end{align*} Or in other words, \begin{align*} \alpha_4 &= \sqrt{5}\\ \alpha_3 &= \sqrt{-32 \alpha_4 - 160}\\ \alpha_2 &= \sqrt{-15118284881920000 \alpha_4 + 34359738368000000}\\ \alpha_1 &= \sqrt[5]{1/512 (1875 \alpha_4 + 3125) \alpha_3 \alpha_2 + 2560000000} \end{align*} up to choosing the correct roots of unity when taking these roots.To express the roots of $f$ in terms of these generators, we enter
[K!r : r in roots];
, which yieldsYou can try this code for yourself using the Magma online calculator.