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If $K$ is an infinite field, and $K(x)$ is the field of rational function (in the indeterminate $x$), is it true that the extension $K(x)/K$ is a Galois extension?

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up vote 4 down vote accepted

Since $x$ is not algebraic over $K$, $K(x)/K$ is not an algebraic extension, hence not a Galois extension.

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