examples such that (i)$F[x]$ is not a field, (ii) $F[x]$ is also a field. [duplicate]

Question

Let $F$ be a field.Then we know that $F[x]$ is a Euclidean Domain.But can someone give me few examples such that (i)$F[x]$ is not a field, (ii) $F[x]$ is also a field.

Thanks for your kind help.

Answer 1

$F[x]$ is never a field, if $F$ is; to see this, try and find a multiplicative inverse for $x$ in $F[x]$; such would have to be a polynomial $p(x) \in F[x]$ with $xp(x) = 1$; but $xp(x)$ has no term of degree zero, whereas $1$ has only a term of degree zero.QED.

The above provides a rich source of examples for $F[x]$ not a field! ;-)!!!

Hope this helps. Cheerio,

and as always,

Fiat Lux!!!

Answer 2

The polynomial ring F[X] is never a field. Consider the element X. What is it's inverse? It doesn't have one in F[X] (i.e. 1/X isn't in F[X]) and so F[X] can't be field.