Lüroth's theorem: Difference between revisions
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==Statement== | ==Statement== | ||
Let <math>K</math> be a field, and <math>K(t)</math> be the [[field of univariate rational functions]], i.e., the field of rational functions in one variable over <math>K</math>. Then, any subfield of <math>K(t)</math> containing <math>K</math> is of the form <math>K(f(t))</math>, where <math>f</math> is a rational function. In particular, this subfield is again the field of univariate rational functions, with the variable now being <math>f(t)</math>. | Let <math>K</math> be a field, and <math>K(t)</math> be the [[field of univariate rational functions]], i.e., the field of rational functions in one variable over <math>K</math>. Then, any subfield of <math>K(t)</math> ''properly'' containing <math>K</math> is of the form <math>K(f(t))</math>, where <math>f</math> is a rational function. In particular, this subfield is again the field of univariate rational functions, with the variable now being <math>f(t)</math>. | ||
In other words, any nontrivial sub-extension of a simple transcendental extension is again a simple transcendental extension. | |||
==Related facts== | |||
* [[Castelnuovo's theorem]] | |||
Latest revision as of 23:07, 14 May 2009
Statement
Let be a field, and be the field of univariate rational functions, i.e., the field of rational functions in one variable over . Then, any subfield of properly containing is of the form , where is a rational function. In particular, this subfield is again the field of univariate rational functions, with the variable now being .
In other words, any nontrivial sub-extension of a simple transcendental extension is again a simple transcendental extension.