Lüroth's Theorem

A theorem that can be stated either in the language of abstract algebraic curves or transcendental extensions.

For an abstract algebraic curve, if x and y are nonconstant rational functions of a parameter, the curve so defined has curve genus 0. Furthermore, x and y may be expressed rationally in terms of a parameter which is rational in them (Coolidge 1959, p. 246).

For simple transcendental extensions, all proper extensions of a field F which are contained in a simple transcendental extension of F are also simple transcendental. In particular, if K is an intermediate field between F and the field F(x) of rational functions over F, then K=F(g(x)) for some nonconstant rational function g(x) (van der Waerden 1966, p. 198).

See also

Transcendental Extension

Portions of this entry contributed by Margherita Barile

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Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 246, 1959.van der Waerden, B. L. Modern Algebra, Vol. 1, 2nd ed. New York: Frederick Ungar, p. 198, 1966.

Referenced on Wolfram|Alpha

Lüroth's Theorem

Cite this as:

Barile, Margherita and Weisstein, Eric W. "Lüroth's Theorem." From MathWorld--A Wolfram Web Resource.

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