Lagmental Vicfred

Conjugates Are Images under Field Embeddings by Vicfred

Last updated: Fri 29 May 2015

tags: mathematics, field extensions, minimal polynomials, degrees

The K-conjugates of a separable algebraic element are its possible images under embeddings into an algebraic closure. This is a compact note, but the quantifiers and hypotheses stay on the page.

The data

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.

$$ m_{\alpha,K}(x)=\prod_{i=1}^{d}(x-\alpha_i) $$

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.

$$ \operatorname{Hom}_K(K(\alpha),\overline K)\longleftrightarrow\{\alpha_1,\ldots,\alpha_d\} $$

Derivation

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

$$ \begin{array}{c|cc}\alpha&\sigma_1(\alpha)&\sigma_2(\alpha)\\\hline\sqrt d&\sqrt d&-\sqrt d\end{array} $$

Invariant content

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad m_{\alpha,K}(x)=\prod_{i=1}^{d}(x-\alpha_i),\\[5pt] \mathsf{C}\;&:\quad \operatorname{Hom}_K(K(\alpha),\overline K)\longleftrightarrow\{\alpha_1,\ldots,\alpha_d\}. \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Hom}_K(K(\alpha),\overline K)\longleftrightarrow\{\alpha_1,\ldots,\alpha_d\} \end{gathered}} $$

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 Sat 01 September 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.