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I am confused in the approach to prove that an specific curve is rational. I know that it means that it is birationally equivalent to $P^1$ but when im working with concrete examples i get confused.

For example,if we take a 1 parameter family (which we will denote by $u$) of curves defined over a field K(u) with $K$ perfect, lets say: $$ y^2 = x^7 u + 1,$$ im not sure how can i prove if it is indeed a rational curve or not. I cannot find a birational map between the curve and $P^1$. Is it easier to work with the function fields? I mean can i write $$K(u)[x,y]/(y^2-x^7 u -1) \cong K\left(\frac{y^2+1}{x^7}\right)[x,y]$$ and prove that it is $K(x,y)$.

Anye help will be apreciated.

Sammy samuel
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  • Where did your $u$ go? Don't you want the field of fractions of the ring $A = K(u)[x,y] / (y^2 - x^7u - 1)$ to be $Q(A) = K(u,t)$ for some $t \in A$? – red_trumpet Jan 08 '24 at 07:51

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