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2022-10-01

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

Cameron Wallace

Beginner2022-10-02Added 8 answers

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 $A(x)=B(x)={x}^{2}+3$. Then $A(x)B(x)=({x}^{2}+3{)}^{2}$ and this polynomial also splits completely over $\mathbb{Q}(\sqrt{-3})$. So the Galois group of $A(x)B(x)$ is still ${C}_{2}\ncong {C}_{2}\times {C}_{2}$

For the composition consider $A(x)=x-3$. Then $A(B(x))={x}^{2}$ has trivial Galois group, but B has Galois group ${C}_{2}$

For the composition consider $A(x)=x-3$. Then $A(B(x))={x}^{2}$ has trivial Galois group, but B has Galois group ${C}_{2}$

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