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I need help to prove that system of intermediate fields of extension $\mathbb{k}(x,y)$, of field $\mathbb{k}(x^p,y^p) \ char(\mathbb{k}) = p > 0$ is infinite

personalfebus
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  • Are you familiar with the theorem that a field extension has infinitely many intermediate fields iff it is not simple? – Thorgott Dec 12 '20 at 15:02
  • See also here or here. I am reluctant to be the first to vote to close as a duplicate because A) I answered those and B) I have the dupehammer privilege for two tags so my vote would be immediately binding. If you agree that this is a duplicate, @-ping me, and I will act. The answer by reuns is, of course, useful. – Jyrki Lahtonen Dec 12 '20 at 16:03
  • Personalfebus, I recommend that you take a look at our guide for new askers. Taking those pointers to heart will improve your use of the site in that your questions will avoid negative attention. – Jyrki Lahtonen Dec 12 '20 at 16:04
  • @JyrkiLahtonen Sorry, I tried to find already existing thead before creating new question, but somehow missed the one you linked. I like both answers provided by reuns here and your answer in another thread. I don't really know whether to mark my question as a duplicate or to it solved with reuns answer. Also I'll look into new askers guide, thank you! – personalfebus Dec 12 '20 at 16:41
  • Don't worry about it too much. You are a new user. I'm more concerned about the users who have spent many years on the site. Then again, this is not like one of those standard calculus exercises. This has not been asked dozens of times already. It was easier for me because I only needed to search within my own answers. Nevertheless, searching is something that you should get into habit of doing before asking. – Jyrki Lahtonen Dec 12 '20 at 16:45

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For $m\ne n$ if $$k(x^p,y^p,x^{mp+1} +y)= k(x^p,y^p,x^{np+1} +y)$$ then $$k(x^p,y^p,x^{mp+1} +y,x^{np+1} +y)=k(x,y)$$

has degree $p$ over $k(x^p,y^p)$ which is a contradiction.

reuns
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