LEC ALG3 19

[!Theorem]
Let $F$ be any field. Then any $f(x)\in F[x]$ of positive degree has a splitting field.

[!Proof]-

Induct on $\text{deg}(f)$. For $\text{deg}(f)=1$, $f(x)=x-a$, where $a\in F$, and $f(x)=0$ when $x=a$. If $\text{deg}f⩾2$, suppose $f$ is irreducible.

Put $k_1=F[x]/(f(x))$. Let the image $x+(f(x))$ of $x$ be denote by ${\alpha }_1$. Note that ${\alpha }_1$ is a root of $f(x)$ in $k_1$. Write $f(x)=(x-{\alpha }_1)f_1(x)$ where $f_1(x)\in k_1[x]$. Write $f_1(x)=(x-\alpha )g_1(x{)}^{m_1}\dots g_s(x{)}^{m_s}$, where each $g$ is irreducible. Use induction hypothesis to find splitting fields $k_1[{\beta }_{i,j},\dots ,{\beta }_{i,j^{\prime }}]$ for each $g_i$. Take their composition.

[!Example]
$f(x)=x^3-2\in \mathbb{Q}[x]$. Cosntruct a splitting field.