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How can I show that $\mathbb{Q}(t,\sqrt{t^3-t})$ is not purely transcendental over $\mathbb{Q}$ I tried assuming that it is purely transcendental and since its transcendence degree is $1$, then it is generated by a single element. Further we have $E=\mathbb{Q}(x)$ for some $x \in E$, but I am unable to reach to a contradiction. Can anyone help me with this?

I read a solution on MSE, but couldn't understand as it was fast paced.

Edit: I am able to do this, see if anyone can help me complete the argument.

Suppose $E=\mathbb{Q}(x)$ where $x\in E$, then $t,\sqrt{t^3-t} \in E$ implies the existence of rational functions $f,g$ such that $t=f(x)$ and $\sqrt{t^3-t}=g(x)$. Hence $$g^2(x)=f^3(x)-f(x)$$ Further note that we also have $x \in E=\mathbb{Q}(t,\sqrt{t^3-t})=\mathbb{Q}(f(x),g(x))$ and hence $x=h(f(x),g(x))$ where $h$ is a rational function in two variables. But I am unable to proceed from here.

permutation_matrix
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  • Can you link the solution – For the love of maths May 24 '21 at 15:04
  • https://math.stackexchange.com/questions/5278/why-is-mathbbqt-sqrtt3-t-not-a-purely-transcendental-extension-of-m – permutation_matrix May 24 '21 at 17:19
  • All the solutions necessarily reflect some special property of the cubic $t^3-t$ (or the elliptic curve $y^2=x^3-x$). This is becase the claim is false, if we replace it with a cubic that has a double root. I would use some facts about algebraic function fields and/or algebraic geometry. I'm sure an argument "just" involving polynomials exists, but may be a bit delicate. What surrounding material is there in your source prior to this exercise? – Jyrki Lahtonen May 25 '21 at 05:34
  • @JyrkiLahtonen, I have covered Galois Theory from Serge Lang's Algebra. Now I am trying some problems. – permutation_matrix May 25 '21 at 08:54
  • @JyrkiLahtonen, I am focusing on the polynomials $f$ and $g$ and the fact that $x$ is a rational function of $f$ and $g$. I am now trying to show that there cannot be any such polynomials satisying $g^2=f^3-f$. But I am not able to conclude – permutation_matrix May 25 '21 at 08:56

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