Steinitz theorem: Difference between revisions

Latest revision as of 16:34, 12 May 2008

Statement

Suppose $k$ is a field and $K$ is a field containing $k$ . Then, there exists a subset $T$ of $K$ , such that:

$T$ is an algebraically independent subset of $K$ ; in other words the field of fractions $k (T)$ embeds inside $K$
$K$ is algebraic over $k (T)$

Since the extension $k (T) / k$ is purely transcendental and the extension $K / k (T)$ is algebraic, Steinitz theorem can be reformulated as: every field extension can be expressed as an algebraic extension of a purely transcendental extension

Related facts

It is important to note that the order of purely transcendental and algebraic matters: not every field extension can be expressed as a purely transcendental extension of an algebraic extension. In fact, there are examples of extensions where the base field is relatively algebraically closed in the extension, but the extension is not purely transcendental.

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 Since the extension <math>k(T)/k</math> is [[purely transcendental field extension|purely transcendental]] and the extension <math>K/k(T)</math> is [[algebraic extension|algebraic]], Steinitz theorem can be reformulated as: ''every field extension can be expressed as an algebraic extension of a purely transcendental extension''
+==Related facts==
+It is important to note that the order of ''purely transcendental'' and ''algebraic'' matters: not every field extension can be expressed as a purely transcendental extension of an algebraic extension. In fact, there are examples of extensions where the base field is [[relatively algebraically closed subfield|relatively algebraically closed]] in the extension, but the extension is not purely transcendental.