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If $K$ is a ring consisting only of zero, then $K[x]$ is a field (edit: from the comments below I learned that it's not). Are there another rings with this property? I think no. If $R$ contains $1 \neq 0$, then $1 \cdot x \in R[x]$ has no inverse because a degree of a product of two polynomials is a sum of degrees of the factors.

How do you think?

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    See [this questio$n$](http://math.stackexchange.com/questions/2514/why-cant-the-polynomial-ring-be-a-field)2012-02-05

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Yes, you are right.

$R[X]=\{\sum_{i=1}^na_iX^i| a_i \in R ~~\text{and}~~ n \in \mathbb N\}$

Let's call the elements in $R[X]$ a polynomial in $X$. A polynomial in $X$ cannot have negative/ fractional exponents in $X$, by definition.

And, as pointed out in the comments, even if $R=\{0\}$, the ring $R[X]$ is not a field. That amounts to saying, there is no field with a single element. Read the comments above for more enlightening dose of information.