For which $f in mathbbC[x,y]$, $fy$ is a field generator of $mathbbC(x,y)$?
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Let $f=f(x,y) in mathbbC[x,y]$.
Call an element $F in mathbbC[x,y]$ a field generator of $mathbbC(x,y)$, if there exists $G in mathbbC(x,y)$ such that $mathbbC(F,G)=mathbbC(x,y)$. If $G$ happens to be in $mathbbC[x,y]$, then call $F$ a good field generator. See, for example, this paper.
Is it possible to characterize all $f in mathbbC[x,y]$ such that $fy$ is a field generator?
Examples:
(1) If $f in mathbbC[x]$, then $fy$ is a (good) field generator, since $mathbbC(fy,x)=mathbbC(x,y)$.
(2) If $f=x+y$, then $fy$ is a good field generator, since $mathbbC(fy,y)=mathbbC(xy+y^2,y)=mathbbC(xy,y)=mathbbC(x,y)$.
Any hints and comments are welcome!
algebraic-geometry polynomials field-theory extension-field irreducible-polynomials
asked Aug 12 at 3:48
user237522
1,8141617
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up vote
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Let $f=f(x,y) in mathbbC[x,y]$.
Call an element $F in mathbbC[x,y]$ a field generator of $mathbbC(x,y)$, if there exists $G in mathbbC(x,y)$ such that $mathbbC(F,G)=mathbbC(x,y)$. If $G$ happens to be in $mathbbC[x,y]$, then call $F$ a good field generator. See, for example, this paper.
Is it possible to characterize all $f in mathbbC[x,y]$ such that $fy$ is a field generator?
Examples:
(1) If $f in mathbbC[x]$, then $fy$ is a (good) field generator, since $mathbbC(fy,x)=mathbbC(x,y)$.
(2) If $f=x+y$, then $fy$ is a good field generator, since $mathbbC(fy,y)=mathbbC(xy+y^2,y)=mathbbC(xy,y)=mathbbC(x,y)$.
Any hints and comments are welcome!
algebraic-geometry polynomials field-theory extension-field irreducible-polynomials
asked Aug 12 at 3:48
user237522
1,8141617
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $f=f(x,y) in mathbbC[x,y]$.
Call an element $F in mathbbC[x,y]$ a field generator of $mathbbC(x,y)$, if there exists $G in mathbbC(x,y)$ such that $mathbbC(F,G)=mathbbC(x,y)$. If $G$ happens to be in $mathbbC[x,y]$, then call $F$ a good field generator. See, for example, this paper.
Is it possible to characterize all $f in mathbbC[x,y]$ such that $fy$ is a field generator?
Examples:
(1) If $f in mathbbC[x]$, then $fy$ is a (good) field generator, since $mathbbC(fy,x)=mathbbC(x,y)$.
(2) If $f=x+y$, then $fy$ is a good field generator, since $mathbbC(fy,y)=mathbbC(xy+y^2,y)=mathbbC(xy,y)=mathbbC(x,y)$.
Any hints and comments are welcome!
algebraic-geometry polynomials field-theory extension-field irreducible-polynomials
asked Aug 12 at 3:48
user237522
1,8141617
Let $f=f(x,y) in mathbbC[x,y]$.
Call an element $F in mathbbC[x,y]$ a field generator of $mathbbC(x,y)$, if there exists $G in mathbbC(x,y)$ such that $mathbbC(F,G)=mathbbC(x,y)$. If $G$ happens to be in $mathbbC[x,y]$, then call $F$ a good field generator. See, for example, this paper.
Is it possible to characterize all $f in mathbbC[x,y]$ such that $fy$ is a field generator?
Examples:
(1) If $f in mathbbC[x]$, then $fy$ is a (good) field generator, since $mathbbC(fy,x)=mathbbC(x,y)$.
(2) If $f=x+y$, then $fy$ is a good field generator, since $mathbbC(fy,y)=mathbbC(xy+y^2,y)=mathbbC(xy,y)=mathbbC(x,y)$.
Any hints and comments are welcome!
algebraic-geometry polynomials field-theory extension-field irreducible-polynomials
asked Aug 12 at 3:48
user237522
1,8141617
edited Aug 12 at 5:03
asked Aug 12 at 3:48
user237522
1,8141617
asked Aug 12 at 3:48
user237522
1,8141617
asked Aug 12 at 3:48
user237522
1,8141617
1,8141617
add a comment |Â
add a comment |Â
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