Let f(x) ∈ F[x] be a monic polynomial of degree n having all its irreducible factors separable over F. Let K ≤ F̅ be the splitting field of f(x) over F, and suppose that f(x) factors in K[x] into

Let

the product (Δ(ƒ))² is the discriminant of f(x). 

a. Show that Δ(f) = 0 if and only if f(x) has as a factor the square of some irreducible polynomial in F[x]. 

b. Show that (Δ(ƒ))² ∈ F. 

c. G(K/F) may be viewed as a subgroup of S̅_n, where S̅_n is the group of all permutations of {α_i| i = 1, · · ·n}. Show that G(K/F), when viewed in this fashion, is a subgroup of A̅_n, the group formed by all even permutations of {α¡ | i = 1, · · ·, n}, if and only if Δ(ƒ) ∈ F. 

Transcribed Image Text:

}7 Πα - α;). i=1