# Questions tagged [splitting-field]

The splitting field of a polynomial with coefficients in a field, $F$, is the smallest field extension to $F$ such that the polynomial decomposes into linear factors. Often used with (galois-theory).

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### Is there an irreducible quartic over $\mathbb{Q}$ whose splitting field is not radical?

It is well known that there are degree 3 irreducible polynomials over $\mathbb Q$ whose splitting fields are not radical. Indeed, there are examples where the splitting field is a real Galois ...
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Exercise of Rotmann Abstract Algebra: If $E/k$ and $E′/k$ are splitting fields of $f(x) \in k[x]$ and there is a radical extension $K_t/k$ with $E ⊆ K_t$, prove that there is a radical extension $K_{r′... 1answer 59 views ### How to prove that$\mathbb{Q}(2\cos40, 2\cos80, 2 \cos160)=\mathbb{Q}(\cos(40))$How to prove that$\mathbb{Q}(2\cos40, 2\cos80, 2 \cos160)=\mathbb{Q}(\cos(40))$(Sorry for using degrees instead of radians) I got that$\mathbb{Q}\cos(40) \leq \mathbb{Q}(2\cos40, 2\cos80, 2 \cos160)...
Let $F$ be a field of characteristic not 2. Let $a,b \in F$ and $K/F$ is a splitting field of the polynomial $(X^{2}-a)(X^{2}-b)$.Prove that the Galois group of $K/F$ is isomorphic to either $\{e\}$ ...
### Which real numbers are contained in $\mathbb{Q}(\cos40+i\sin40)$?
I have a field extension $\mathbb{Q}(\cos40+i\sin40)$. I would like to find all the real numbers in this field and prove that they form a subfield (S) and find $|S:\mathbb{Q}|$. And also calculate the ...