I have some trouble understanding this proof (companion matrices). Watch

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My attachments are (in order):

a) Corollary 2.130
b) Lemma 8.32
c) Lemma 8.34
d) Theorem 8.5 (i) (the one that I'm having trouble with)
e) The proof of theorem 8.5 (i) that uses a), b), and c) to prove d).

1) For the part that says "By Lemma 8.34, each W_i has a basis B_i...", I'm a bit confused because...

"V" in the lemma corresponds to "W_i" in this proof, right? Then we also need to assume that there exists W^T that is a cyclic k[x]-module. How is that possible?

2) For the part that says "...and Proposition 8.32 shows that T has the desired matrix...". I'm not sure about what they mean by "the desired matrix".

3) Corollary 2.130 tell us that we can write y[T]y as x[T]x by multiplying the right side by P^{-1} and the left by P, right? But in the proof, what are they mulitplying A with in order to get C(g_1) \bigoplus ... \bigoplus C(g_t)?
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