1
vote
1answer
43 views

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 ...
1
vote
1answer
65 views

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, ...
1
vote
1answer
72 views

$\sqrt[7]{11}$ 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 $x^7-12$ ...
0
votes
1answer
140 views

How to solve this by galois theory?

please focus on the concept to solve this problem, because i can't handle to research on diffcult terminology.Thanks in advance. Find all real roots by galois theory and find all other root to this ...
7
votes
2answers
156 views

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 ...