Answer to Question #134315 in Abstract Algebra for Aayushi Deshmukh

Question #134315
Show that polynomial x⁵=9x+3 Is not solvable by radicals over.
1
Expert's answer
2020-09-21T16:20:50-0400
"Solution"

We need to show that the Galois group is not solvable.

Firstly, we note that "f" is irreducible by Eisenstein’s criterion with "p= 3". Therefore, the zeros off are all conjugate, so the Galois group acts transitively on them. Next we observe that "f(x)" has 3 real roots and two complex roots. We can see that "f(\u2212\u221e) =-\u221e,f(0) = -3,f(1) =\u221211" , and "f(\u221e) =\u221e" , so by the intermediate value theorem, we see there are at least 3 real zeros. On the other hand, the roots can’t all be real, since the sum of their squares is 0.


Alternatively, we use some basic calculus to see that "f^\u2032(x) = 5x^4\u22129" , so the only turning points for fare zeros of "x^4\u2212\\frac95" , which has only 2 real solutions.Therefore, "f" has exactly 3 real solutions and two complex ones.


Therefore its Galois group is a transitive subgroup of "S_5" , which contains a transposition, so it must be the whole of "S_5", which is not solvable, so "f" is not solvable by radicals over Q.




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