Find the field F and the polynomial $\displaystyle f(x)=x^2+ax+b$ from F[x] with the derivation identically equal to 0, f(x) is not the second power of another polynomial.
Find the field F and the polynomial $\displaystyle f(x)=x^2+ax+b$ from F[x] with the derivation identically equal to 0, f(x) is not the second power of another polynomial.
Can anybody give me a hint please?
Thanks
such a field has to be an infinite field of characteristic $\displaystyle 2$ and we must also have $\displaystyle a=0.$ here's an example that satisfies the conditions in your problem:
Spoiler:
$\displaystyle F=\mathbb{F}_2(t),$ the field of rational functions over $\displaystyle \mathbb{F}_2,$ and $\displaystyle f(x)=x^2+t.$