- #26

Office_Shredder

Staff Emeritus

Science Advisor

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[K(x):K]=[K(x):K(x

^{2})][K(x

^{2}):K]

We know [K(x):K(x

^{2})] has to be odd, because [K(x):K] is. But [K(x):K(x

^{2})] cannot be more than 2 (because we found x is a root of a quadratic polynomial). So it's either 2 or 1, and can't be 2, so must be one. But a degree one extension means that the two fields are the same