# Tagged Questions

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### Galois Group of $x^n - a$

Homework problem: If the field F contains a primitive nth root of unity, prove that the Galois group of $x^n - a$, for $a \in F$, is abelian. I'm not really sure where to start here and I'm ...
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### How do we explain the existence of complex conjugation?

We can define the complex numbers by writing $\mathbb{C} = \mathbb{R}[i]/(i^2+1),$ where $\mathbb{R}$ is to be regarded as a commutative ring. Furthermore, since $\mathbb{R}$ happens to be a field, ...
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### $\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$

I want to prove: $\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$. Is there any direct way to prove? I have computed that the splitting field of $x^7-12$ ...
### Invariant under transformation $i\mapsto -i$ implies real?
When one has an expression in terms of $i$, one can send $i$ to $-i$ and, if the expression remains unchanged, one can conclude that the expression is, in fact, real. Analogous statements hold for ...