Abstract Algebra: Minimal Polynomial help. Just want to double check my process.

Let a=sqrt(1-sqrt(3))
Find the minimal polynomial for a over Q,R,C.

I am already sure of my answer over Q, but just want to make sure I've done R, C correctly.

Over R,

let x=sqrt(1-sqrt(3))
x^2=1-sqrt(3)
x^2-1+sqrt(3) is my final polynomial, the discriminant is

Question

Abstract Algebra: Minimal Polynomial help. Just want to double check my process.

Let a=sqrt(1-sqrt(3))
Find the minimal polynomial for a over Q,R,C.

I am already sure of my answer over Q, but just want to make sure I've done R, C correctly.

Over R,

let x=sqrt(1-sqrt(3))
x^2=1-sqrt(3)
x^2-1+sqrt(3) is my final polynomial, the discriminant is <0, so it is irreducible.

For C,
let x=sqrt(1-sqrt(3))
x-sqrt(1-sqrt(3)) is my final polynomial, it is linear so that's why I think it works, but just not sure...

anyone know for sure?