MATH 3231 Class Notes 03-07-2025

notefield-theory

Defined Normal Extension

Recall Splitting Field

Theorem 1

If $[E:k]=d<\infty$, ${}^E{/}_k$ normal, then $\exists P(x)\in k[x]$ such that

$${}^E{/}_k\simeq {\mathrm{split}}_k(P(x))(\text{an isomorphism of extensions}).$$

Proof

$[E:k]=d<\infty ⟹\exists {\beta }_1,\dots ,{\beta }_d$ a basis for ${}^E{/}_k$ as a vector space over $k$.

Let $m_i(x)\stackrel{def}{=}{m}_{{\beta }_1/k}(x)$ $P(x)={\prod }_{i=1}^d{m}_i(x)$.

Claim ${}^E{/}_k$ isa. splitting field for $P(x)$ over $k$. $⋮$

Theorem 2

If $k$ field, $P(x)\in k[x]$ nonzero, $E={\mathrm{split}}_k(P(X))$ is a finite normal extension of $k$.

Note

Not all $\alpha \in E$ are roots of $P(x)$, so statement is not trivial. Why does $m_{\alpha /k}(x)$ necessarily split as a product of $\mathrm{deg}1$ polynomials in $E[x]$?

\usepackage{tikz-cd}
 
\begin{document}
 
\begin{tikzcd} && k &&& k \\ 
\\ 
&&&&&&& {\phi(A(\alpha))A(\beta)\ \forall A[x] \in k[x] } 
\\ && {k[\alpha]} &&& {k[\beta]} && {\phi(\alpha)=\beta} 
\\ 
\\ 
E \\ 
&& F &&& F 
\arrow["{\operatorname{id}_k}", tail reversed, from=1-3, to=1-6] 
\arrow[hook, from=1-3, to=4-3] 
\arrow[hook, from=1-6, to=4-6] 
\arrow["\sim", from=4-3, to=4-6] 
\arrow["\circlearrowleft", shift left=5, curve={height=-30pt}, draw=none, from=4-3, to=4-6] 
\arrow["\circlearrowleft"', shift right=5, curve={height=30pt}, draw=none, from=4-3, to=4-6] 
\arrow[hook, from=4-3, to=6-1] 
\arrow[hook, from=4-3, to=7-3] 
\arrow[hook, from=4-6, to=7-6] 
\arrow[hook, from=6-1, to=7-3] 
\arrow["{\exists \varphi \text{ isom.}}", from=7-3, to=7-6] 
\end{tikzcd}
 
\end{document}

Remark

We know how to construct many elements of $\mathrm{Gal}({}^{{\mathrm{split}}_k(P(x))}{/}_k)$.

Example

Compute $\mathrm{Gal}({\mathrm{split}}_{\mathbb{Q}}({}^{(x^3-2)}{/}_{\mathbb{Q}}))$.

Solution: $x^3-2=(x-\sqrt[3]{2})(x-\sqrt[3]{2}w)(x-\sqrt[3]{2}{w}^2)$. If $\phi \in \mathrm{Gal}(\mathbb{Q}[\sqrt[3]{2}=\alpha ,\sqrt[3]{2}w={\alpha }_2,\sqrt[3]{2}{w}^2={\alpha }_3]/\mathbb{Q})$. $\phi$ sends $\left\{\alpha ,{\alpha }_2,{\alpha }_3\right\}$ to $\left\{\alpha ,{\alpha }_2,{\alpha }_3\right\}$ (sends roots to conjugates). $\phi$ injective (automorphism), $\phi$ permutes $\alpha ,{\alpha }_2,{\alpha }_3$.