Dummit And Foote Solutions Chapter 14 ((free))

Computing the groups for specific types of polynomials (e.g., quadratics, cubics, and cyclotomic polynomials).

Let $K$ be a field of characteristic $p > 0$ and let $f(x) \in K[x]$ be a polynomial of degree $n$. Show that the Galois group of $f(x)$ over $K$ has order dividing $n!$. Dummit And Foote Solutions Chapter 14

, you primarily only need to worry about normality (splitting fields). Use the tower rule to determine the size of the Galois group. Computing the groups for specific types of polynomials (e

A subfield $E$ is Galois over $\mathbbQ$ iff the corresponding subgroup $H$ is normal in $G$. $1, \sigma^2$ is normal (center of $D_8$), so $\mathbbQ(\sqrt2, i)$ is Galois (indeed, it's a compositum of quadratic extensions). $1, \tau$ is not normal (conjugate to $1, \sigma^2\tau$), so $\mathbbQ(\sqrt[4]2)$ is not Galois over $\mathbbQ$ (it doesn’t contain $i\sqrt[4]2$). , you primarily only need to worry about