Galois Group of $x^2-25$ over $\mathbb{Q}$

Question

I had seen a question asking to determine the Galois group of the polynomial $f(x)=x^2-25$ over $\mathbb{Q}$. Clearly this reduces to $f(x)=(x-5)(x+5)$. However, I don't understand how we are suppose to determine the Galois group here. The roots are $R(f)=\{±5\}$. Which are already contained in $\mathbb{Q}$. Hence there is no need to construct a splitting field. Am I missing something here?

Answer 1

The polynomial given splits over $\mathbb{Q}$, and the Galois group of a polynomial over a given ground field are the field automorphisms of the polynomial's splitting field which act as the identity on the given ground field. Our ground field is the rationals, and the splitting field is also the rationals. Now, can you describe field automorphisms of the rational numbers that also act as the identity on the rational numbers? These such maps will serve as the elements of your desire Galois group.