Section 3.11 for September 24

1. (Difficult) I understood most of the information about fields, but why does x+1 = x^2 in the group Z_2[X]? I know that it was a law in the first example, but I do not understand where it comes from. The last page was a little fuzzy for me once it went into recurrence. I understand how to set up the matrix and most of the steps and explanation, but I don't know what Lagrange's theorem in group theory is, so I didn't understand exactly how the value 2^m -1 comes into play.

2. (Reflective) I have seen most of this stuff in Abstract Algebra. I remember doing a short lesson and homework assignment on irreducible polynomials and feeling really confused, but I think that I understand it better now. It makes more sense now that we have been doing some homework problems using binary numbers, and the example in the book used Z_2[X]. I still do not understand all of the theory behind groups and fields, but it is becoming clearer every time I study it a little more. It is definitely a different kind of mindset to think of certain polynomials being irreducible or not irreducible mod 2, when that is not always the case when working with real numbers.