26 questions linked to/from Why can't the Polynomial Ring be a Field?
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### If $F$ is a field show that $F[x]$ is not a field. [duplicate]

I know that $ax=1$ has a solution in $F$ so that every element must be a unit but then I'm not sure how to proceed.
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### Let $F$ be a field. Could the ring $F[x]$ be a field? [duplicate]

$F$ 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 ...
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### Is $R[x]$ never a field? [duplicate]

I know that if $F[x]$ is a PID then $F$ is a field. Now $F[x]$ being a field implies that $F[x]$ is a PID, so $F$ is a field. Anyway, I tried to prove that $F$ is a field right away and the following ...
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### examples such that (i)$F[x]$ is not a field, (ii) $F[x]$ is also a field. [duplicate]

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.
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### Disproving that $\mathbb R[x]$ is a field [duplicate]

The task is to determine if $\mathbb R[x]$, which represents the set of all polynomials with real coefficients, is a field. My response is that it is not, since I feel that not all of these ...
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### 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 $\mathbb{Q}[x]$ ...
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### gcd of $x$ and $2$ in $\mathbb Z[x]$

In $\mathbb Z[x]$, $x$ and $2$ have gcd $1$. But they cannot be expressed as the linear combination of two polynomials. Then assuming that $1=2f(x)+xg(x)$ we are supposed to arrive at a ...
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### Subfields of Rings

I am currently working through an undergraduate class in Galois Theory. I have come across a question that I am unsure about. Can a ring that is not a field, have a subring that satisfies the ...
I have a homework problem which says If $f(x) = \dfrac{x^2 - x}{x - 1}$ and $g(x) = x$. Is it true that $f = g$? What do they mean by saying is it true that $f = g$? Aren't these two functions ...
### Why isn't $\frac{1}{x}$ a polynomial?
Why isn't $\frac{1}{x}$ a polynomial? Does it directly follow from definition? As far as I know, polynomials in $F$ are expressions of the form $\sum_{i=0}^{n} a_ix^i$, where $a_i\in F$ and $x$ is a ...