Polynomial division algorithm and finding inverse matrix?

Question

Algorithms

Hello, still torturing this Reed Solomon code. Already more or less figured out, but there are gaps. Help please)
In general, I said a pdf file with a description of this thing. And some things are not clear at all. For example:
1) Here
is a photo where polynomials are divided. I don’t understand how they divide like that, for example, we write 9x^8 and multiply by the divisor, we will get 9x^14 + 63x^13 + ... And they have 9x^14 + 10x^13 + ... (Ps 63 is probably taken modulo, but I don’t understand which one?In the textbook there is no information about the Galois field, exactly about the number of elements for this example ..), and I can’t find the modulus coefficient so that everything else converges, can you please tell me
2) In the example
By some miracle, the inverse matrix was found. I generally got it with fractions (on the Internet I drove an online solution and there are fractions, even negative numbers)
How can I find this matrix? A little higher in the same document they wrote that the inverse matrix is ​​searched as in the usual way ...

Answer 1

https://cyberleninka.ru/article/v/delenie-mnogochl...