Lagmental Vicfred

The Tower Formula Multiplies Extension Degrees by Vicfred

Last updated: Sun 10 September 2023

tags: mathematics, field extensions, minimal polynomials, degrees

Finite dimensions in a chain of fields multiply exactly as dimensions of nested vector spaces. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Statement

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.

$$ K\subseteq F\subseteq L $$

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.

$$ [L:K]=[L:F][F:K] $$

Worked algebra

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \left.\begin{aligned}\{u_i\}_{i=1}^{m}&\text{ basis of }F/K,\\\{v_j\}_{j=1}^{n}&\text{ basis of }L/F\end{aligned}\right\}\Longrightarrow\{u_iv_j\}_{i,j}\text{ basis of }L/K $$

Conceptual compression

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad K\subseteq F\subseteq L,\\[5pt] \mathsf{C}\;&:\quad [L:K]=[L:F][F:K]. \end{aligned} $$

Caveat

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] [L:K]=[L:F][F:K] \end{gathered}} $$

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 Mon 18 May 2015. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.