The only difficulty I had with this reading is seeing the importance of some of the latter theorems concerning whether F is a subring of F[x]/(p(x)), because my initial understanding of the concepts renders the theorem obvious since the constant polynomials in F[x] will still be distinct congruence classes and if you were only to look at their multiplication addition tables, you would have the same thing as F. It'd be nice to know if I'm missing something here...
In reflecting on the material, it appears that we can construct F[x]/(p(x)) to be a field itself provided p(x) is reducible, because then the gcd of p(x) and all the members of congruence classes will be 1.
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