Negative exponents in Lang's algebra exercise 56

[stupiduser]

Hi,

I have been puzzled by this exercise for some time. I won't repeat it here as the hint refers to other exercises, so I would be copying a whole page of the book.

My question is as follows. After having defined the sets $$S_i$$ and picked a complex number $$s$$ not belonging to any of the $$S_i$$ it is trivial to use part (b) of exercise 55 to show that for some large $$k$$ we have $$M_i^k(s)\in S_i$$ for all $$i$$.

Now the problem is that we still need to prove that $$M_i^{nk}(s)\in S_i$$ for all $$n\neq 0$$ i.e. all the exponents, but I don't see how this can be done for negative exponents.

Further, I have no idea how to prove that there exists such a $$k$$ that $$M_i^k(S_j)\subset S_i$$ for all $$j\neq i$$ and further for all the exponents of $$M_i^k$$. It doesn't even seem plausible that this could happen as any element close to $$w'_i$$ is (for a large enough $$k$$) mapped into $$U_i$$ i.e. close to $$w_i$$ by part (b) of the previous exercise. Thus, it would seem that choosing some $$z\in U_i'$$ closer and closer to $$w_i'$$ would require $$k$$ to grove without bound...

I'm just wondering whether anyone has ever done this exercise or has some tips on how to approach it?

Thanks.

 

[stupiduser]

Nevermind the first part. I just realized that the inverse of a matrix has the same fixed points, so we get the required condition for the element s for negative exponents. This doesn't help with the second problem relating the sets though.