A Splitting Field Is Generated by Every Root

The splitting field of a polynomial is the smallest extension in which it factors completely into linear terms. This is a compact note, but the quantifiers and hypotheses stay on the page.

Statement

A finite extension $L/K$ is Galois when it is both normal and separable. Its group $G=\operatorname{Gal}(L/K)$ records all automorphisms fixing $K$.

$$ f(x)=a\prod_{i=1}^{n}(x-\alpha_i) $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in $\mathcal A$ may change ambient meaning, so I keep $\mathsf D$ separate from $\mathsf C$.

$$ \operatorname{Split}_K(f)=K(\alpha_1,\ldots,\alpha_n) $$

Worked algebra

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ x^3-2=(x-\sqrt[3]2)(x-\zeta_3\sqrt[3]2)(x-\zeta_3^2\sqrt[3]2),\qquad L=\mathbf Q(\sqrt[3]2,\zeta_3) $$

Conceptual compression

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad f(x)=a\prod_{i=1}^{n}(x-\alpha_i),\\[5pt] \mathsf{C}\;&:\quad \operatorname{Split}_K(f)=K(\alpha_1,\ldots,\alpha_n). \end{aligned} $$

Caveat

Normal and separable are independent hypotheses outside perfect fields. Having the right degree alone does not make an extension Galois.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Split}_K(f)=K(\alpha_1,\ldots,\alpha_n) \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Sun 26 July 2020. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.

A Splitting Field Is Generated by Every Root by Vicfred

Statement

Worked algebra

Conceptual compression

Caveat