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$F$ is a field, so by definition, $F$ is a commutative ring with unity in which every non-zero element is a unit. Then, $F[x]$ is a set of polynomials in which the coefficients come from $F$, so all of the non-zero coefficients have units in $F$.

This means we can have polynomials like $f=a_0+a_1x+a_2x^2+\cdots +a_nx^n$ where $a_k\in F,$ $0\le k\le n$.

If $F[x]$ were a field, every non-zero element would be a unit.

I'm not really sure where to go from here.

TheMobiusLoops
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  • I don't think so. – TheMobiusLoops Oct 27 '13 at 02:33
  • It would be $n+1$. – TheMobiusLoops Oct 27 '13 at 02:35
  • So it can't have inverses because if it did, the degree would be 0. Right? – TheMobiusLoops Oct 27 '13 at 02:37
  • Geometrically, taking a ring $R$ and adding an indeterminate $R[x]$ is like "adding a dimension" to $R$. All fields have dimension $0$, and the above shows that for any ring (non-zero) $R$, $R[x]$ has dimension at least $1$. I could make this rigorous, but I don't want to--you already have a good rigorous answer, this is just intuition carries over to many settings. For example, you could ask if $\mathbb{Z}$ could ever be isomorphic to $F[x,y]$ for a field $F$, and the same intuition works. – Alex Youcis Oct 27 '13 at 03:16

2 Answers2

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You can "turn it into a field" by taking the quotient ring $F[x]/M$ for a maximal ideal $M$ of $F[x]$. This is sometimes called sending $M$ to zero.

This is actually another structure which is indeed a field, but it's not related isomorphically to $F$. So a better way to describe this is inducing a field using $F[x]$ and $M$.

Daniel Donnelly
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$x$ is not zero in $F[x]$ and has no multiplicative inverse. So $F[x]$ cannot be a field.

Rob Arthan
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