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user237522
user237522
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Characterizing all simple algebraic ring extensionextensions of $\mathbb{C}[x]$ having no prime elements

Let $w$ be an algebraic element over $k[x]$$\mathbb{C}[x]$, with minimal polynomial $f(t)=c_mt^m+\cdots+c_1t+c_0$, $c_i \in k[x]$$c_i \in \mathbb{C}[x]$.

Is it possible to characterize (in terms of the $c_j$'s) all algebraic elements $w$, such that $R=k[x][w]$$R=\mathbb{C}[x][w]$ has no prime elements?

Please see this related question, in which the special case $k[x^2][x^3]$ is dealt with (notice that in this special case, $m=2$ with $c_2=1$, namely, $w$ is integral).

Characterizing all simple algebraic ring extension of $\mathbb{C}[x]$ having no prime elements

Let $w$ be an algebraic element over $k[x]$, with minimal polynomial $f(t)=c_mt^m+\cdots+c_1t+c_0$, $c_i \in k[x]$.

Is it possible to characterize (in terms of the $c_j$'s) all algebraic elements $w$, such that $R=k[x][w]$ has no prime elements?

Please see this related question, in which the special case $k[x^2][x^3]$ is dealt with (notice that in this special case, $m=2$ with $c_2=1$, namely, $w$ is integral).

Characterizing all simple algebraic ring extensions of $\mathbb{C}[x]$ having no prime elements

Let $w$ be an algebraic element over $\mathbb{C}[x]$, with minimal polynomial $f(t)=c_mt^m+\cdots+c_1t+c_0$, $c_i \in \mathbb{C}[x]$.

Is it possible to characterize (in terms of the $c_j$'s) all algebraic elements $w$, such that $R=\mathbb{C}[x][w]$ has no prime elements?

Please see this related question, in which the special case $k[x^2][x^3]$ is dealt with (notice that in this special case, $m=2$ with $c_2=1$, namely, $w$ is integral).

the title had mismatched braces i fixed them
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edit approved May 11, 2017 at 14:52
David Richter
edit approved May 11, 2017 at 14:52
David Richter
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Characterizing all simple algebraic ring extension of $\mathbb{C][x]$C}[x]$ having no prime elements

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user237522
user237522
  • 2.8k
  • 14
  • 24

Characterizing all simple algebraic ring extension of $\mathbb{C][x]$ having no prime elements

Let $w$ be an algebraic element over $k[x]$, with minimal polynomial $f(t)=c_mt^m+\cdots+c_1t+c_0$, $c_i \in k[x]$.

Is it possible to characterize (in terms of the $c_j$'s) all algebraic elements $w$, such that $R=k[x][w]$ has no prime elements?

Please see this related question, in which the special case $k[x^2][x^3]$ is dealt with (notice that in this special case, $m=2$ with $c_2=1$, namely, $w$ is integral).