We have elements a in the ring R=Q[x]/(x^5+x^4-x-1)
If a is nilpotent, find the smallest positive k with a^k=0
If a is not nilpotent, but a zero divisor, find a non-zero be that is smallest so that ab=0
If it is not a zero divisor, find its inverse.
I don't think any are nilpotent...I've tried various ks to no avail, but I could be wrong
i) I got a zero divisor with b=x^2-1
I think ii) has an inverse since x^5+x^4-x-1=0, then x^5+x^4-x=1, so that x(x^4+x^3-1)=1, and so it has inverse x^4+x^3-1.
iii) I'm not sure here
iv) I got a zero divisor with b=x+1
Do these look right, or am I missing someway any of these are nilpotents, and what about iii)? Thank you.