Exercise 9.3.6 (Polynomial unsolvable by radicals)

Question

Prove that for every $n \ge 5$, there is some polynomial of degree $n$ that is not solvable by radicals.

Hassaan Naeem · Accepted Answer

extbf{Solution:}
By 	extbf{Theorem 9.3.5} we have that not every polynomial of degree 5 is solvable by radicals, let one such polynomial be $f(t)$. 
For $n>5$, assume for sake of contradiction, that every polynomial of degree $n$ is solvable by radicals.
We would then have that $f(t)\cdot t^{n-5}$ is solvable by radicals, which would imply that $f(t)$ is solvable
by radicals, hence a contradiction.

Exercise 9.3.6 (Polynomial unsolvable by radicals)

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Exercise 9.3.6 (Polynomial unsolvable by radicals)

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