If f is a polynomial with one variable, then I want to prove that $y^3+f(x)$ can be written as the product of polynomials with positive degree if and only if there is a polynomial g(x) such that $f(x)=g(x)^3$, one direction is simple, if $f(x)=g(x)^3$ then $y^3+f(x)=(y+f(x))(y^2-yf(x)+f(x)^2)$, but how to prove the another direction? I have no clue, can anyone help me? thank you in advance, and also please help me in this question:$f_{n}\left ( x,y \right )+f_{n-1}\left ( x,y \right )$ is irreducible for two homogeneous polynomials without common divisor. thank you
Asked
Active
Viewed 38 times
1
-
what is $y$? A constant? – miracle173 Sep 13 '21 at 09:32
-
@miracle173 y is a variable – YuerCauchy Sep 13 '21 at 09:33
-
I think there's no difference between a constant an a variable. But it is independent of $x$. Is this true? – miracle173 Sep 13 '21 at 09:36
-
yes @miracle173 – YuerCauchy Sep 13 '21 at 09:37
1 Answers
5
If $y^3+f$ is the product of two polynomials then these can polynomial factors be written as $y+a$ and $y^2+by+c$ where $a,b,c$ are polynomials in $x$.
So $$(y+a)(y^2+by+c)=y^3+y^2(a+b)+y(ab+c)+ac=y^3+f$$
comparing the coefficients we get
$$a+b=0\\ ab+c=0\\ ac=f $$ and further $b=-a$, $c=a^2$ and $f=a^3$
miracle173
- 11,049