The second field is the function field of $X_2:=E\times {\mathbb P}^1$, where $E$ is a smooth elliptic curve; the first one is the function field of $X_2:={zy^2-x^3-tz^2=0}\subset X_1:={zy^2-x^3-tz^2=0}\subset {\mathbb P}^3$. The surface $X_2$ X_1$ is rational, as one can see by projecting onto ${\mathbb P}^2$ from the point $P$ given by $x=y=z=0$, which is a double point of $X_2$. X_1$. The surface $X_1$ X_2$ is not rational, since it has $h^1({\mathcal O}_{X_1})=1$O}_{X_2})=1$. So $X_1$ and $X_2$ are not birational, and the two fields are not isomorphic (as extensions of $\mathbb C$).
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The second field is the function field of $X_2:=E\times {\mathbb P}^1$, where $E$ is a smooth elliptic curve; the first one is the function field of $X_2:={zy^2-x^3-tz^2=0}\subset {\mathbb P}^3$. The surface $X_2$ is rational, as one can see by projecting onto ${\mathbb P}^2$ from the point $P$ given by $x=y=z=0$, which is a double point of $X_2$. The surface $X_1$ is not rational, since it has $h^1({\mathcal O}_{X_1})=1$. So $X_1$ and $X_2$ are not birational, and the two fields are not isomorphic (as extensions of $\mathbb C$). |
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