Dummit And Foote Solutions Chapter 14

Understanding the relationship between fields and their automorphism groups. Galois Groups: Computing Galois groups for specific polynomial extensions. Fundamental Theorem of Galois Theory:

Let $w \in W$ and $g \in G$. Since $W$ is $G$-invariant, we have $g \cdot w \in W$. Applying $\rho(g)$, we get $\rho(g)w \in W$, which shows that $\rho(G)W \subseteq W$. Dummit And Foote Solutions Chapter 14