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 $K$.

Solution: The minimal polynomial of $\zeta_5$ over $\mathbb{Q}$ is the $5$th cyclotomic polynomial $\Phi_5(x) = x^4 + x^3 + x^2 + x + 1$. Since $\Phi_5(x)$ is irreducible over $\mathbb{Q}$ (by Eisenstein's criterion with $p = 5$), we have $[\mathbb{Q}(\zeta_5):\mathbb{Q}] = 4$. The roots of $\Phi_5(x)$ are $\zeta_5, \zeta_5^2, \zeta_5^3, \zeta_5^4$, and $\mathbb{Q}(\zeta_5)$ contains all these roots. Hence, $\mathbb{Q}(\zeta_5)/\mathbb{Q}$ is a splitting field of $\Phi_5(x)$ and therefore a Galois extension. abstract algebra dummit and foote solutions chapter 4

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: ($\Rightarrow$) Suppose $f(x)$ splits in $K$

You're looking for solutions to Chapter 4 of "Abstract Algebra" by David S. Dummit and Richard M. Foote! showing that $f(x)$ splits in $K$.

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.

($\Leftarrow$) 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 $K$.