isomorphic to Galois group of the polynomial $(x-1)^2(x-3)^3(x-5)$

Question

please some help to find the group that isomorphic to Galois group of the polynomial $f(x) = (x-1)^2(x-3)^3(x-5)$

Answer 1

Since $f$ splits over $\mathbb Q$, the Galois group of $f$ is the group of automorphisms of $\mathbb Q$ fixing $\mathbb Q$. What is that group?

Edit: A polynomial $f(x)$ is said to split over a field if it breaks into linear factors over that field.

For example, the polynomial $f(x) = x^2 + 1$ does not split over $\mathbb Q$ or $\mathbb R$ but it does split over $\mathbb Q(i)$, and $\mathbb C$. The polynomial $(x^2-2)(x^2-3)$ does not split over $\mathbb Q$ or $\mathbb Q(\sqrt 2)$ or $\mathbb Q(\sqrt 3)$, but it does split over $\mathbb Q(\sqrt2,\sqrt3)$. (Why?)

An extension of a field $F$ is just any larger field in which $F$ embeds. For example, $\mathbb C$ is an extension of $\mathbb R$ and $\mathbb R$ is an extension of $\mathbb Q$.