I am trying to determine the splitting fields of a bunch of polynomials. I'll ask one here and hope that a general enough technique can be described to find the rest of them.
Currently, I'm trying to find the splitting field of $(x^{15}-5)(x^{77}-1)$ over $\Bbb Q$, find the degree, and determine if it's a Galois extension.
Now, I know that the right polynomial is the cyclotomic polynomial, hence has degree $\varphi(77)=60$, over $\Bbb Q$. The left polynomial is irreducible by Eisenstein's Criterion, hence adjoining $\sqrt[15]{5}$ gives a degree 15 extension and as a separate extension, adjoining $\zeta_{15}$ (a primitive $15^{th}$ root of unity) gives a degree $8$ extension. Since $8$ and $15$ are relatively prime, I know that the degree of the extension for the splitting field of $x^{15}-5$ is $120$.
All of this seems well and good, but now I'm lost. The splitting field itself is obviously $\Bbb Q(\zeta_{77},\sqrt[15]{5},\zeta_{15})$, but how can I check the degree and determine if it is Galois?
Best Answer
First reduce the $\zeta$'s: you have $$\mathbb{Q}(\zeta_{77}, \zeta_{15}) = \mathbb{Q}(\zeta_{lcm(77,15)}) = \mathbb{Q}(\zeta_{1155}).$$ This is an extension of $\mathbb{Q}$ of degree $\varphi(1155) = \varphi(77)\cdot \varphi(15) = 480.$
What is the intersection $\mathbb{Q}(\zeta_{1155}) \cap \mathbb{Q}(\sqrt[15]{5})$?
Subfields of cyclotomic fields are abelian (the converse is also true), that is, they are Galois with abelian Galois groups. However, the nontrivial subfields of $\mathbb{Q}(\sqrt[15]{5})$ - $\mathbb{Q}(\sqrt[3]{5})$, $\mathbb{Q}(\sqrt[5]{5})$ and $\mathbb{Q}(\sqrt[15]{5})$ - are not Galois. So you have $\mathbb{Q}(\zeta_{1155}) \cap \mathbb{Q}(\sqrt[15]{5}) = \mathbb{Q}$ and $$[\mathbb{Q}(\zeta_{1155}, \sqrt[15]{5}) : \mathbb{Q}] = 480 \cdot 15 = 7200.$$