Chapter 4 - Abstract Algebra Dummit And Foote Solutions

Exercise 4.2.2: Let $K$ be a field, $f(x) \in K[x]$, and $L/K$ a splitting field of $f(x)$. Show that $L/K$ is a finite extension.

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 $f(x)$. 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 $K$. Exercise 4

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 $a + b, ab, a^{-1} \in K^G$. Since $f(x)$ is separable, there exists $\sigma \in

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 $K^{\times}$.