1. find the field

Find the field F and the polynomial $f(x)=x^2+ax+b$ 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

2. Originally Posted by sidi
Find the field F and the polynomial $f(x)=x^2+ax+b$ 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 $2$ and we must also have $a=0.$ here's an example that satisfies the conditions in your problem:

Spoiler:
$F=\mathbb{F}_2(t),$ the field of rational functions over $\mathbb{F}_2,$ and $f(x)=x^2+t.$