I am in my first abstract algebra course and was working on a seemingly innocent exercise in my textbook that turned out to be much more difficult than I imagined.
"Show $x^2 - 1$ has two distinct factorizations into irreducibles in Z8"
It's easy to find two factorizations and easy to show they are distinct.
Those factors are $x + 1$ , $x + 3$ , $x + 5$ and $x + 7$
As I'm not in a field (or even in a domain!) many of my normal strategies fail.
One thing I have noticed and hope will help is that the coefficients of all of my linear polynomials are units in Z8.
In the specific case $x + 1$, if we assume it is the product of two non unit polynomials, then I force the constant terms of the two non unit polynomial factors to be the same.
Thank you so much!