Zero divisors and such
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.
Okay, I think I get it.
I am just a little confused about parts of it...
I got i, ii....for iii, you are saying that the inverse of x^2 would have to be (x^4+x^3-1)^2? Also, I do not see what k would have to be to get a^k=0.
I see how it could be a zero divisor by long division, but not sure about this.
a*b=1, then (a*b)*(a*b)=1. Multiplication is commutative here, so a^2*b^2=1. There's no reason that b^2 can't be the multiplicative inverse of a^2.
To compute the desired k, remember, in ring R, for all . See which k can make g(x) really a polynomial on Q rather than a rational function?