Let $K$ be a field and $D = K[X]$. I need to show that if $f\in D$ is non constant, then the extension of rings $K[f]\subset D$ is integral, and if $A$ is a subring of $D$ which contains $K$ and has Krull dimension $1$, then $A$ is a finitely generated $K$-algebra.

Thank you in any advance.

  • 1
    $\begingroup$ Uh? You wrote " if f is in D but not in D..." ?? $\endgroup$ – DonAntonio Jun 15 '13 at 0:51
  • $\begingroup$ the question was edited, I think its possible reopen this now... Thank you...@DonAntonio, @Clayton, $\endgroup$ – User43029 Jun 17 '13 at 14:42
  • $\begingroup$ What have you tried so far? You can assume wlog that $f$ is monic. Now you can write down explicitly a polynomial which shows that $X$ is integral over $K[f]$. etc. $\endgroup$ – Martin Brandenburg Jun 18 '13 at 13:31
  • $\begingroup$ For the second part of your question can find an answer here. $\endgroup$ – user26857 Jun 18 '13 at 16:53

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