Mixing
A small exercise on polynomial rings
$begingroup$
Let $k$ be a field of characteristic zero and let $S=k[x,y,z]$. Suppose that $f,gin S$ are such that $x$ divides $yf + zg$. Can we conclude that $x$ divides $f$ and, of course, $g$? Any hint?
Thank you!
polynomials
asked Jan 15 at 18:57
FranciscoFrancisco
786
$endgroup$
add a comment |
$begingroup$
Let $k$ be a field of characteristic zero and let $S=k[x,y,z]$. Suppose that $f,gin S$ are such that $x$ divides $yf + zg$. Can we conclude that $x$ divides $f$ and, of course, $g$? Any hint?
Thank you!
polynomials
asked Jan 15 at 18:57
FranciscoFrancisco
786
$endgroup$
2
$begingroup$
Let $f=z$ and $g=-y$, then $yf+zg=0$ is divisible by $x$, but neither $f$ nor $g$.
$endgroup$
– Qurultay
Jan 15 at 19:21
$begingroup$
Of course, thank you.
$endgroup$
– Francisco
Jan 15 at 20:03
add a comment |
$begingroup$
Let $k$ be a field of characteristic zero and let $S=k[x,y,z]$. Suppose that $f,gin S$ are such that $x$ divides $yf + zg$. Can we conclude that $x$ divides $f$ and, of course, $g$? Any hint?
Thank you!
polynomials
asked Jan 15 at 18:57
FranciscoFrancisco
786
$endgroup$
Let $k$ be a field of characteristic zero and let $S=k[x,y,z]$. Suppose that $f,gin S$ are such that $x$ divides $yf + zg$. Can we conclude that $x$ divides $f$ and, of course, $g$? Any hint?
Thank you!
polynomials
polynomials
asked Jan 15 at 18:57
FranciscoFrancisco
786
asked Jan 15 at 18:57
FranciscoFrancisco
786
asked Jan 15 at 18:57
FranciscoFrancisco
786
asked Jan 15 at 18:57
FranciscoFrancisco
786
asked Jan 15 at 18:57
FranciscoFrancisco
786
786
2
$begingroup$
Let $f=z$ and $g=-y$, then $yf+zg=0$ is divisible by $x$, but neither $f$ nor $g$.
$endgroup$
– Qurultay
Jan 15 at 19:21
$begingroup$
Of course, thank you.
$endgroup$
– Francisco
Jan 15 at 20:03
add a comment |
2
$begingroup$
Let $f=z$ and $g=-y$, then $yf+zg=0$ is divisible by $x$, but neither $f$ nor $g$.
$endgroup$
– Qurultay
Jan 15 at 19:21
$begingroup$
Of course, thank you.
$endgroup$
– Francisco
Jan 15 at 20:03
2
2
$begingroup$
Let $f=z$ and $g=-y$, then $yf+zg=0$ is divisible by $x$, but neither $f$ nor $g$.
$endgroup$
– Qurultay
Jan 15 at 19:21
$begingroup$
Let $f=z$ and $g=-y$, then $yf+zg=0$ is divisible by $x$, but neither $f$ nor $g$.
$endgroup$
– Qurultay
Jan 15 at 19:21
$begingroup$
Of course, thank you.
$endgroup$
– Francisco
Jan 15 at 20:03
$begingroup$
Of course, thank you.
$endgroup$
– Francisco
Jan 15 at 20:03
add a comment |
0
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2
$begingroup$
Let $f=z$ and $g=-y$, then $yf+zg=0$ is divisible by $x$, but neither $f$ nor $g$.
$endgroup$
– Qurultay
Jan 15 at 19:21
$begingroup$
Of course, thank you.
$endgroup$
– Francisco
Jan 15 at 20:03