Prove-or-disprove-For-any-extension-E-of-a-field-F-F-u-F-u-u-E-Where-F-u-is-a-smallest-subfield-of-E-containing-F-and-u-and-F-u-f-u-f-x-F-x-F-x-is-a-polynomial-ring-over-F-

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Question Number 169989 by udaythool last updated on 13/May/22

$$$$$$\mathrm{Prove}\mathrm{or}\mathrm{disprove}:$$$$\mathrm{For}\mathrm{any}\mathrm{extension}E\mathrm{of}\mathrm{a}\mathrm{field}F,$$$$F\left(u\right)=F\left[u\right]\mathrm{\forall }u\in E.$$$$\mathrm{Where}F\left(u\right)\mathrm{is}\mathrm{a}\mathrm{smallest}\mathrm{subfield}$$$$\mathrm{of}E\mathrm{containing}F\mathrm{and}u\mathrm{and}$$$$F\left[u\right]=\{f\left(u\right)\mid f\left(x\right)\in F\left[x\right]\},F\left[x\right]\mathrm{is}\mathrm{a}$$$$\mathrm{polynomial}\mathrm{ring}\mathrm{over}F.$$

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