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We know that $\mathbb{Q} [x,y]/(x)$ is isomorphic to $\mathbb{Q}[y]$.

My question is that

Is $\mathbb{Q}[y]$ is field ? Yes/No

My attempt: I think yes, because we know $ \mathbb{Q}[x] $ is a field and an integral domain in a similar way we can also said that $\mathbb{Q}[y]$ is a field and an integral domain

Is its true ?

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    $\begingroup$ $\mathbb{Q}[x]$ is not a field: $x$ has no multiplicative inverse. $\endgroup$ – lhf Aug 27 '19 at 18:29
  • $\begingroup$ oksss @Ihf i mean $\mathbb{Q}$ is field i thinks im not getting the concept $\endgroup$ – jasmine Aug 27 '19 at 18:33
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    $\begingroup$ Since you know that $\mathbb{Q} [x,y]/(x)$, can you show that $(x)$ is not maximal? Can you think about any larger ideal? $\endgroup$ – N. S. Aug 27 '19 at 18:34
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    $\begingroup$ You may confuse $\mathbb Q[x]$ with $\mathbb Q[\alpha]$ where $\alpha$ is an algebraic number.... $\endgroup$ – N. S. Aug 27 '19 at 18:34
  • $\begingroup$ exactly im confused with $\mathbb{Q} [\alpha ]$@N.S. i think $ (x,y) $ will be the maximal ideal of $\mathbb{Q}[x,y]$ $\endgroup$ – jasmine Aug 27 '19 at 18:37