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‘nt to the rest of the world.
Blah blah blah, the balloon falls, blah blah blah.
Blah blah blah, the old guy I meet on the plane goes to the bathroom, I go to the lady’s room, she claims she’s pregnant, I guess I’ll go back to my seat and rest, and of course I take a nap, and soon I’m jolted awake as the sky’s about to explode (let’s just say that I know what the earth looks like through an airplane window).
.
Nt to the rest of the world, they say, “Oh, you’re just a world-famous.
Blah blah blah, the grumpy stewardess names me a tardy passenger, but I don’t care because I’m about to board a spaceship to the future, blah blah blah.
Blah blah blah, I get sucked into the time vortex, blah blah blah.
Blah blah blah, I’m talking to the last person on Earth, blah blah blah.
“Let’s say you can time travel.
Blah blah blah, I go back a day, but it’s a different day, blah blah blah.
Blah blah blah, I meet up with myself, blah blah blah.
Blah blah blah, I meet up with myself (again), blah blah blah.
Blah blah blah, I get carried away in my own world, blah blah blah.
Blah blah blah, I get carried away with my own world, blah blah blah.
Blah blah blah, I find a secret door, blah blah blah.
Blah blah blah, the door can transport you anywhere, blah blah blah.
Blah blah blah, I go to a secret door, blah blah blah.
Blah blah blah, the door leads me back a day, blah blah blah.
Blah blah blah, I’m on the moon, blah blah blah.
Blah blah blah, I meet up with myself (again), blah blah blah.
Blah blah blah, we reunite, blah blah blah.
Blah blah blah, I reveal that I’m from a different timeline, blah blah blah.
Blah blah blah, I reveal that I’m from a different timeline, blah blah blah.
Blah blah blah, I go to a secret door, blah blah
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Changing of the homolomorphism $varphi colon text{Gal}(mathbb{Q}[x]/mathbb{Q}[x]P) to text{Gal}(mathbb{Q}(x)/mathbb{Q})$
Let $P in mathbb{Q}[x]$ be a monic polynomial with $P(0)
e 0$ and $P in mathbb{Q}[x] setminus mathbb{Q}$ i.e. the zero locus of $P$ is not a subset of $mathbb{Q}$.
Consider the homomorphism of $varphi colon text{Gal}(mathbb{Q}[x]/mathbb{Q}[x]P) to text{Gal}(mathbb{Q}(x)/mathbb{Q})$ with the definition
begin{equation}
varphi colon text{Gal}(mathbb{Q}[x]/mathbb{Q}[x]P) to text{Gal}(mathbb{Q}(x)/mathbb{Q}), quad sigma mapsto bar{sigma}
end{equation}
where $bar{sigma}$ is the extension of $sigma$ to the algebraic closure of $mathbb{Q}$.
The question is:
Can I change the homomorphism $varphi$ such that the image of $varphi$ becomes the Galois group of
$mathbb{Q}[x]/mathbb{Q}[x]P(x)$?
I know the Galois group $text{Gal}(math
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