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Let $p$ be a prime number, and $k = \Bbb{F}_p(t)$ be a function field over $\Bbb{F}_p$. Let $R = k[x,y]/(x^p+y^p-t)$, and $K = \operatorname{QF}(R)$ be the quotient field of $R$. I need to find the transcendence degree of $K/k$, but my issue is unwinding all the definitions to get where I want.

I know $R$ is a UFD, which may or may not be useful. I attempted to write out an arbitrary element of $K$ but it was super messy, and probably not useful. It seems that $x$ and $y$ are transcendental elements over $k$ in $K$, but the algebraic relationship $x^p+y^p -t$ is throwing me off. It's a two-variable polynomial expression so that $x$ and $y$ should still be algebraically independent in $K$, I think.

So, my question is how do I seek out a transcendence basis for $K$ over $k$?

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Hint: $k \subset k(x) \subset K$ and has transcendence degree $1$. On the other hand, $y\in K$ is a root of $p(Y) =Y^p + (x^p-t) \in k(x)[Y]$. The algebric relation is what makes then algebraically dependent.

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  • $\begingroup$ So for a transcendence basis of $K$ over $k$, given the ideal relationship, we only need $\{x\}$ or $\{y\}$, but not both? $\endgroup$ – walkar Apr 9 '16 at 18:27
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    $\begingroup$ Like in the linear case, in order to be basis as set has to be algebraically independent, so if $x,y$ belong to a transendence basis there are no polynomials relations (over k) $p(x,y)=0. $\endgroup$ – Vinicius M. Apr 10 '16 at 11:44
  • $\begingroup$ Thanks, I think I have it. I needed it to use some facts about Kahler differentials, and the field theory was tripping me up. $\endgroup$ – walkar Apr 10 '16 at 14:27

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