# What is the galois group of x+3 or (x+1)(x+2) ? How about A(x)B(x)?

What is the galois group of $x+3$ or $\left(x+1\right)\left(x+2\right)$ ? How about $A\left(x\right)B\left(x\right)$ ?
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Cameron Wallace
The idea that the Galois group of a product of polynomials is the direct product of the groups is false, however (assuming that is what you mean). Consider $A\left(x\right)=B\left(x\right)={x}^{2}+3$. Then $A\left(x\right)B\left(x\right)=\left({x}^{2}+3{\right)}^{2}$ and this polynomial also splits completely over $\mathbb{Q}\left(\sqrt{-3}\right)$. So the Galois group of $A\left(x\right)B\left(x\right)$ is still ${C}_{2}\ncong {C}_{2}×{C}_{2}$
For the composition consider $A\left(x\right)=x-3$. Then $A\left(B\left(x\right)\right)={x}^{2}$ has trivial Galois group, but B has Galois group ${C}_{2}$