Abstract Algebra Dummit And Foote Solutions Chapter 4 -

Solution: Let $\alpha_1, \ldots, \alpha_n$ be the roots of $f(x)$. Then $L = K(\alpha_1, \ldots, \alpha_n)$, and $[L:K] \leq [K(\alpha_1):K] \cdots [K(\alpha_1, \ldots, \alpha_n):K(\alpha_1, \ldots, \alpha_{n-1})]$.

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 $K^{\times}$.

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. abstract algebra dummit and foote solutions chapter 4

Solution: ($\Rightarrow$) Suppose $f(x)$ splits in $K$. Then $f(x) = (x - \alpha_1) \cdots (x - \alpha_n)$ for some $\alpha_1, \ldots, \alpha_n \in K$. Hence, every root of $f(x)$ is in $K$.

You're looking for solutions to Chapter 4 of "Abstract Algebra" by David S. Dummit and Richard M. Foote! Solution: Let $\alpha_1, \ldots, \alpha_n$ be the roots

($\Leftarrow$) 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 $K$.

Exercise 4.1.2: Let $K$ be a field and $G$ a subgroup of $\operatorname{Aut}(K)$. Show that $K^G = {a \in K \mid \sigma(a) = a \text{ for all } \sigma \in G}$ is a subfield of $K$. If $a = 0$, then $\sigma(a) = 0$

Solution: Let $\alpha$ and $\beta$ be roots of $f(x)$. Since $f(x)$ is separable, there exists $\sigma \in \operatorname{Aut}(K(\alpha, \beta)/K)$ such that $\sigma(\alpha) = \beta$. By the Fundamental Theorem of Galois Theory, $\sigma$ corresponds to an element of the Galois group of $f(x)$, which therefore acts transitively on the roots of $f(x)$.

Exercise 4.3.2: Let $K$ be a field and $f(x) \in K[x]$ a separable polynomial. Show that the Galois group of $f(x)$ acts transitively on the roots of $f(x)$.