4.1

1. What was the most difficult part of the reading for you?
The proof of the division algorithm, definitely. I'm sure it's just because it's kind of late and my brain isn't fully functioning anymore, but I'm going to have to come back to that 2-page proof another day. It makes logical sense to me, but that thing is pretty intense. I also think that reading/understanding it WILL be important, but it sure is scaring me right now. Similarly, the use of "F[x]" Didn't make the most sense in the world to me.

2. Explore F[x]
Okay, I'll start by restating what the book says about F[x]: ...x is given a meaning similar to "[x sometimes doesn't seem to stand for any particular number but is treated as if it were a number in simplification exercises]", then the polynomials with coefficients in a field F form a ring (denoted F[x]) whose structure is remarkably similar to that of the ring Z of integers. So, I'll try to restate that: F[x] is a RING of the polynomials with coefficients that came from a different ring (in this case, F). So, and they may have said this already, but they are saying that F is a subring of F[x], right? And the reason F is a subring of F[x] is that F[x] which will have the same elements (the coefficients) of F will also have X in it? Is that right? I feel like that is what I understood from that, but wow, I'm not quite sure. I guess we'll see in class tomorrow and when I try to do the homework for this section...