Show $\Bbb R$ and $\Bbb R[X]$ not isomorphic

Question

Why are the rings $\mathbb{R}$ and $\mathbb{R}[ x ]$ not isomorphic to eachother ?

Think it might have to do with multiplicative inverses but I'm not sure.

However they are isomorphic as additive groups (and even as $\Bbb Q$-vector spaces). — Watson, Aug 19 '16 at 14:12

score 3 · Accepted Answer · answered Apr 02 '15 at 00:22

3

You are right: the element "$x$" has no multiplicative inverse, that is there is no polynomial $p(x)$ such that $x\cdot p(x)=1$.

answered Apr 02 '15 at 00:22

Salomo

1

And there is a slick yet insightful way to prove that $,x,$ is not invertible. – Bill Dubuque Apr 02 '15 at 01:51

score 2 · Answer 2 · answered Apr 02 '15 at 00:23

2

Well $\Bbb R$ is a field while $\Bbb R[X]$ is not.

Question: if a ring is ring-isomorphic to a field, is it necessary a field?

answered Apr 02 '15 at 00:23

Alexandre Halm

2 Answers2