Lagmental Vicfred

A Minimal Polynomial Is the Kernel of Evaluation by Vicfred

Last updated: Mon 02 March 2020

tags: mathematics, field extensions, minimal polynomials, degrees

For an algebraic element, evaluation from K[x] has a principal prime kernel generated by one irreducible polynomial. I want the notation, the mechanism, and the failure mode visible at the same time.

Objects and 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.

$$ \operatorname{ev}_\alpha:K[x]\to K(\alpha),\qquad f\mapsto f(\alpha) $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \ker(\operatorname{ev}_\alpha)=(m_{\alpha,K}) $$

Push the symbols

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ K[x]/(m_{\alpha,K})\xrightarrow{\sim}K[\alpha],\qquad [K(\alpha):K]=\deg m_{\alpha,K} $$

Structural reading

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

$$ \begin{aligned} \mathsf{D}\;&:\quad \operatorname{ev}_\alpha:K[x]\to K(\alpha),\qquad f\mapsto f(\alpha),\\[5pt] \mathsf{C}\;&:\quad \ker(\operatorname{ev}_\alpha)=(m_{\alpha,K}). \end{aligned} $$

A hypothesis worth keeping

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \ker(\operatorname{ev}_\alpha)=(m_{\alpha,K}) \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Mon 02 May 2016. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.