Answer to Question #131865 in Abstract Algebra for Bharti

Question

Show that the polynomial
X^5=9x+3 is not solvable by radicals over

Answer 1

The Galois group of the polynomial x⁵ − 9x - 3 over Q is S₅ Proof. Let f(x) = x⁵ − 9x - 3, we have that f is irreducible over Q by Eisenstein’s Criterion theorem . Since there are two sign changes in the polynomial, then by Descartes’ rule of signs, we have that f(x) has 2 or 0 positive real roots. Since f(0) = -3 and f(1) = −11, then by intermediate value theorem, we have that ∃x ∈ (0, 1) such that f(x) = 0, thus we have that f(x) has at least one positive real zero, then by previous proof, we have that f(x) has 2 positive real roots. By calculation, we have that f(−x) = −x 5 + 9x - 3, and there’s one change of signs, then by corollary to Descartes’ rule of signs, we have that f(x) has 1 real negative zero. By combination, we have that f(x) has 3 real zeros, each has multiplicity 1. Then we know that there are 2 complex conjugating zeros, namely a + bi and a − bi by A non-zero polynomial of degree n over a field has at most n zeros, counting multiplicity theorem . It remains for us to show that the Galois group of f over Q is S₅. Let the five roots of f(x) be a₁, a₂, a₃, a₄, a₅, let K = Q( a₁, a₂, a₃, a₄, a₅,), a K-automorphism that permutes ai , thus we know that Gal(K/Q) is isomorphic to a subgroup of S₅. Since we know that a₁ is a zero of an irreducible polynomial of degree 5, we have that [Q(a₁) : Q] = 5, and 5|[K : Q]. By Fundamental Theorem of Galois Theory , we know that 5||Gal(K/Q)|, then we have that Gal(K/Q) contains an element of order 5. Since Gal(K/Q) is isomorphic to a subgroup of S₅, and Gal(K/Q) has an element of order 5, we know that Gal(K/Q) contains a 5-cycle. We also have that Gal(K/Q) contains a mapping α sending a + bi to a − bi, which fixes the three real roots and permutes the two complex roots, then we have that α is a 2-cycle since it has 2 distinct elements. By A 2-cycle and a 5-cycle generate S₅ theorem , we conclude that a 2-cycle and a 5-cycle generates S₅, thus we have that Gal(K/Q) ∼= S₅, as desired.