Exercise 4.3.1: Show that $\mathbb{Q}(\zeta_5)/\mathbb{Q}$ is a Galois extension, where $\zeta_5$ is a primitive $5$th root of unity.
Exercise 4.2.1: Let $K$ be a field and $f(x) \in K[x]$. Show that $f(x)$ splits in $K$ if and only if every root of $f(x)$ is in $K$. abstract algebra dummit and foote solutions chapter 4
You're looking for solutions to Chapter 4 of "Abstract Algebra" by David S. Dummit and Richard M. Foote! Exercise 4
Solution: Let $a \in K$. If $a = 0$, then $\sigma(a) = 0$. If $a \neq 0$, then $a \in K^{\times}$, and $\sigma(a)$ is determined by its values on $K^{\times}$. You're looking for solutions to Chapter 4 of
Solution: Clearly, $0, 1 \in K^G$. Let $a, b \in K^G$. Then for all $\sigma \in G$, we have $\sigma(a) = a$ and $\sigma(b) = b$. Hence, $\sigma(a + b) = \sigma(a) + \sigma(b) = a + b$, $\sigma(ab) = \sigma(a)\sigma(b) = ab$, and $\sigma(a^{-1}) = \sigma(a)^{-1} = a^{-1}$, showing that $a + b, ab, a^{-1} \in K^G$.
Exercise 4.1.1: Let $K$ be a field and $\sigma$ an automorphism of $K$. Show that $\sigma$ is determined by its values on $K^{\times}$.