Here are some solutions to selected exercises from Chapter 4:
($\Leftarrow$) Suppose every root of $f(x)$ is in $K$. Let $\alpha_1, \ldots, \alpha_n$ be the roots of $f(x)$. Then $f(x) = (x - \alpha_1) \cdots (x - \alpha_n)$, showing that $f(x)$ splits in $K$. abstract algebra dummit and foote solutions chapter 4
Section 4.3 deals with groups acting on themselves by conjugation. This leads to the , a vital tool for counting and understanding the "center" of a group. the sylow theorems and their applications Here are some solutions to selected exercises from