Abstract Algebra: Theory and Applications · Problem 23.5.4

Galois Theory — Thomas Judson

Mathematics — Abstract Algebra · galois-theory-solvability-by-radicals

Problem

Determine the Galois groups of each of the following polynomials in

\mathbb{Q}[x]

; hence, determine the solvability by radicals of each of the polynomials. (a)

x^5 - 12x^2 + 2

(b)

x^5 - 4x^4 + 2x + 2

(c)

x^3 - 5

(d)

x^4 - x^2 - 6

(e)

x^5 + 1

(f)

(x^2 - 2)(x^2 + 2)

(g)

x^8 - 1

(h)

x^8 + 1

(i)

x^4 - 3x^2 - 10