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Unramified Local Extensions Are Controlled by Residue Fields by Vicfred

Last updated: Mon 27 September 2010

tags: mathematics, infinite Galois theory, local fields, ramification

A finite extension of local fields is unramified when its ramification index is one and the residue extension accounts for the degree. I want the notation, the mechanism, and the failure mode visible at the same time.

Notation

Infinite Galois groups carry the Krull topology and become profinite groups. For a discretely valued field \(K\), completions and residue fields add a second layer of arithmetic to extensions \(L/K\).

$$ [L:K]=ef $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ L/K\ \text{unramified}\Longleftrightarrow e=1,\quad f=[L:K] $$

Stress the formula

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \begin{array}{ccc}\mathcal O_L&\twoheadrightarrow&k_L\\\cup&&\cup\\\mathcal O_K&\twoheadrightarrow&k_K\end{array}\qquad\operatorname{Gal}(L/K)\cong\operatorname{Gal}(k_L/k_K) $$

Interpretation

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

$$ \begin{aligned} \mathsf{D}\;&:\quad [L:K]=ef,\\[5pt] \mathsf{C}\;&:\quad L/K\ \text{unramified}\Longleftrightarrow e=1,\quad f=[L:K]. \end{aligned} $$

Limit of the argument

Subgroups in infinite Galois theory correspond to intermediate fields only after taking closure. Ramification filtrations also depend on the chosen valuation.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] L/K\ \text{unramified}\Longleftrightarrow e=1,\quad f=[L:K] \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Sun 17 January 2010. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.