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

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 ?

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