Let F be a field of characteristic Show that f(x) is irreducible over F or f(x) splits in F. p and let f(x) = x² − a € F[x].

Let f be an entire function such that Re(f(z)) > 0 for all z EC. Prove that f is a constant.

Given a function f: ℝ → ℝ^2, for each x ∈ ℝ write f(x) = [f1(x), f2(x)] ∈ ℝ^2.Where that f1 and f2 are real valued functions of ℝ. Prove f is differentiable at a ∈ ℝ, if and only if, f1 and f2 are differentiable at a, and in that case     f'(a)  = [f1'(a)  f2'(a)] (right part of equation should be in column form)     For the following exercises, use the following definition:A function f: ℝ^n → ℝ^m is differentiable at a ∈ ℝ^n if there exists a linear transformationg: ℝ^n → ℝ^m such thatlim [f(a+h)-f(a)-g(h)]/h =0h→0

Find curl F and div F for the following F=(2y-cos(x)) i -z²e3*j+(x²-7z) k