I'm taking rings and fields this semester but don't remember group theory. Is it necessary to review everything? I took group theory a year and a half ago so I don't remember anything. I'm taking rings and fields this semester and I'm worried I won't be prepared (group theory was already a struggle back then). We're covering chapters 12-22 in Gallian's Contemporary Abstract Algebra (7th or 8th edition, they're the same units).
Should I be reviewing everything or can I go on without?
Thank you.
 A: You probably won't need to review quite as much as you might think. Your mileage may vary of course, but here's what I remember from my experience:


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*You should be able to recite the definition of a group (closure, associativity, identity, inverse) in your sleep. Rings and fields are defined in terms of groups, so this will certainly help you in the long run.

*You should review normal subgroups and factor groups. Recall that the factor group $G/H$ only makes sense if $H$ is a normal subgroup of $G$. You'll learn about an analogous result with ideals and factor rings: the factor ring $R/A$ only makes sense if $A$ is an ideal of $R$.

*You should review group homomorphisms, kernels, and the First Isomorphic Theorem for groups. You'll learn a similar result when it comes to ring homomorphisms and the First Isomorphic Theorem for rings.
Other than that, you won't need too much of the other stuff that you learned from group theory. It may actually be helpful to review anything you've learned from number theory. It would be great if you could be comfortable with:


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*applying the Division Algorithm via long division

*finding the GCD

*applying the Extended Euclidean Algorithm

*applying the Chinese Remainder Theorem


The above techniques that work for integers will also come in handy when you later apply them to polynomials instead.
A: I'm posting this around 2 years after I took Rings & Fields. Just in case anyone might find my input helpful, I'll post it.
I actually did quite well and didn't find it too difficult without reviewing my group theory. In fact, I think it was easier than group theory (or maybe it's just that I just matured a bit more in various ways). In a way as well, rings and fields is kind of its own thing; an extension of group theory, but I found it self-contained (at least by Gallian) so one could more or less learn it on its own without involving group theory.
From my personal experience, I didn't find a review of group theory to be necessary, but that's my personal experience.
