As the title says I wonder what the galois group of $x+3$ is. Or even if that exists ? Since $x+3 = 0$ has only one zero/element I assume its the trivial group ? And what is the galois group of $(x+1)(x+2)$ ?
Assume we have (integer) polynomials $A(x)$ and $B(x)$ and we know their galois group as $A'$ and $B'$. Is the galois group of $A(x)B(x)$ the group $A' \times B'$ ?
Is the galois group of $A(B(x))$ the group $A' \times B'$ ?
Are there tricks for products or compositions ?
Sorry I am new to Galois theory. My questions are not random as they might appear. I know what a group is, but I have never seen this addressed specifically, and solving quintics is a bit much for a newby. I know I should read as much as possible, but answering these questions would probably help me get rid of my misunderstandings.