1. The topics that believe will be important to know are roots and irreduciblity as well as some basics facts about a ring mod by an ideal, or a Field mod by an irreducible. I think knowing the idea behind Thrm. 4.5 will be important as well as Thrm 4.11. I think Thrm 6.1 as well as cosets, quotient rings and kernels will all be necessary to know.
2. I need to work on understanding quotient rings, cosets and ideals. I also need to clarify when certain definitions and theorems apply- in other words if you must have a field, a commutative ring, etc. I need to better understand different properties regarding degrees of polynomials. I also need to try and understand the First Isomorphic Thrm. better and understand when to apply it.
3. In the proof of Cor. 4.16 I don't understand why (c-a) can't be equal to the zero in F. The whole strategy of this proof is also confusing for me. In regards to cosets I was wondering if a coset of I in R means any elements of I plus any element of R where the elements in R are related to the representative? If this is true, does this mean that an ideal is always just some set of multiples of something? How does taking an element and adding something from I make it congruent to to something else in R mod I?
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