I am trying to show that the polynomial $y-x^2-x^3$ is reducible in the formal power series ring $k[[x,y]]$. I am attempting the question by finding a polynomial in $k[[x,y]]$ which is the square root of $x^2+x^3$. In order to find the square root I wrote the general polynomial in $k[[x,y]]$, $$a_{00}+a_{10}x+a_{01}y+a_{20}x^2 ...........$$ Took it's square and equated the coefficients to the coefficients of $x^2+x^3$. I got the following system of equations
$$a_{00}^2=0$$ $$2a_{10} a_{00}=0$$ $$2a_{20} a_{00}+{a_{10}}^2=1 $$ and so on. But this system doesnot have a solution. I am sure that the root does exist. What am I doing wrong?
By the generalized binomial theorem we have $$ \sqrt{x^2+x^3}=|x| (1+x)^{\frac{1}{2}}=|x| \sum_{i=0}^\infty \binom{\frac{1}{2}}{i}x^i, $$ where $$ \binom{\frac{1}{2}}{i}=\frac{\frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-2) \cdots (\frac{1}{2}-i+1)}{i!} $$