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Let $k$ be a field of characteristic zero, and let $k[x,x^{-1},y]$ be the polynomial ring in $x,x^{-1},y$.

Is there a Euclidean algorithm in $k[x,x^{-1},y]$?

Two relevant questions are: this and this.

Any hints and comments are welcome!

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A Euclidean domain (i.e. an integral domain with a Euclidean function that allows the Euclidean algorithm to find gcd's, like the absolute value for $\Bbb Z$, or the degree for $k[x]$) must necessarily be a principle ideal domain. The ideal $(x+1, y)\subseteq k[x, x^{-1}, y]$ is not principal. So there is no Euclidean algorithm on your ring.

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  • $\begingroup$ Thank you. Same answer for $k[x,y]$? $\endgroup$ – user237522 Jun 19 at 10:48
  • $\begingroup$ @user237522 Exactly (except there one can use the ideal $(x, y)$, which looks a bit nicer). Imagine using the extended Euclidean algorithm to write $1 = f\cdot x + g\cdot y$. That's clearly not going to work. Your case requires a little more care since $x$ is invertible, but not much. $\endgroup$ – Arthur Jun 19 at 10:50
  • $\begingroup$ Thank you. $k[x,x^{-1},y]$ is a UFD, isn't it? ($k[x,y]$ is a UFD en.wikipedia.org/wiki/Unique_factorization_domain). $\endgroup$ – user237522 Jun 19 at 11:55
  • $\begingroup$ @user237522 Yes, I believe it is. Which in particular means that gcd's exist. You just can't use the Euclidean algorithm to get to them. $\endgroup$ – Arthur Jun 19 at 11:57
  • $\begingroup$ Thank you. (I have also seen this argument). $\endgroup$ – user237522 Jun 19 at 12:04

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