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I'm learning Galois theory and want to solve the following problem.

Let $K=\mathbb{C}(x)$, $f(y)=y^4+xy^2+x\in K[y], L$ is the splitting field of $f$ over $K$. What is the Galois group of $L/K$？

My idea is that if $\alpha$ is a root of $f$, then $-\alpha$ is another root of $f$. If we can find the other two roots of $f$ in terms of $\alpha$, then we can calculate the Galois group. But I have no idea how to calculate.

algebra-learner
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  • $f$ is a quadratic polynomial in $y^2$. So you can contemplate a certain tower of two quadratic extensions... – Qiaochu Yuan Jan 21 '25 at 15:36
  • It seems to me that this post gives you everything you need. My vote to close would take immediate effect, so I will wait for a while. Mostly because there may be an alternative solution based on function field techniques rather than the general result of the duplicate target. – Jyrki Lahtonen Jan 21 '25 at 16:24
  • Closing this for now. If somebody wants to add a solution that cannot equally well go to the duplicate target (may be a live possibility), just at-ping me. – Jyrki Lahtonen Jan 21 '25 at 20:57

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