Lagmental Vicfred

Algebraic Elements Form a Field by Vicfred

tags: mathematics, field extensions, minimal polynomials, degrees

Elements algebraic over K remain algebraic after addition, multiplication, and inversion. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

Notation

An extension \(L/K\) is a vector space together with compatible multiplication. The degree \([L:K]\) is its vector-space dimension, so bases and minimal polynomials control field size.

$$ \overline K_L=\{\alpha\in L:\alpha\text{ algebraic over }K\} $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ \alpha,\beta\in\overline K_L\Longrightarrow\alpha\pm\beta,\ \alpha\beta,\ \alpha^{-1}\in\overline K_L $$

Stress the formula

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

$$ [K(\alpha,\beta):K]\le[K(\alpha):K]\,[K(\beta):K]<\infty $$

Interpretation

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

$$ \begin{aligned} \mathsf{D}\;&:\quad \overline K_L=\{\alpha\in L:\alpha\text{ algebraic over }K\},\\[5pt] \mathsf{C}\;&:\quad \alpha,\beta\in\overline K_L\Longrightarrow\alpha\pm\beta,\ \alpha\beta,\ \alpha^{-1}\in\overline K_L. \end{aligned} $$

Limit of the argument

The tower formula requires finite degrees for ordinary integer multiplication. Infinite extensions need cardinal dimensions or separate algebraic arguments.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \alpha,\beta\in\overline K_L\Longrightarrow\alpha\pm\beta,\ \alpha\beta,\ \alpha^{-1}\in\overline K_L \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Tue 28 February 2023. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.