## Math 350: Groups, Rings and Fields Study Guide for the Final Exam

The final exam is on Tuesday, December 19, from 9:00am to 12:00pm, in our regular classroom SMUD 206.

Reading/exam period office hours are Th 10:00-11:50am, M 12:30-2:00pm.

The following is a chapter by chapter guide intended to help you organize the material we have covered in class as you study for your exam. It is only intended to serve as a guideline, and may not explicitly mention everything that you need to study.

The final exam is cumulative, so it is important that you study the material from earlier chapters as well. Please be sure to review homework and example problems for the chapters given below. If you would like to see additional examples and practice problems, I recommend looking at the textbook A First Course in Abstract Algebra by John B. Fraleigh, which is on reserve for this course in the Science Library.

Another good source of practice problems comes from the department comprehensive exams. You may also find it useful to look over the summary of topics covered in the Algebra section of the comprehensive exam.

16: Know the definition of a rings and those of related terms, such as commutative rings, unity, units, zero-divisors, nilpotent, direct sum, integral domain, division ring and field. Also know the statements and proofs of the theorems and corollaries in this section.

17: Know the definitions of subring and ideal, and be able to check whether a given subset of a ring is either of these. Know what it means for an ideal to be principal, prime, or maximal. Understand how a quotient ring is formed using a ring and an ideal, and what types of quotient rings you get using different types of ideals.

18: Know the definition of a ring homomorphism, and the properties of these homorphisms as given in the statements and proofs of Theorems 18.1-18.5. You do not need to know the Second and Third Isomorphism Theorems, or about prime subfields or characteristics. You should understand and be able to carry out the construction of the field of fractions of a domain, as given in Theorem 18.10.

19: You should know what we mean by R[X] for a given ring R, and be able to verify that this is a ring. You should know the terms degree, leading coefficient, constant polynomial, root, and irreducible. Know the results and proofs of the theorems and corollaries in this chapter up to the end of p.197. You should also be able to use the Eisenstein Criterion (Thm 19.11) to determine whether a polynomial is reducible over the rational numbers or not. You do not need to know the terms content or primitive, or Lemmas 19.9 and 19.10, for this exam.

20: Know the statements and proofs of the theorems in this chapter, and practice working through the examples so that you are able to construct field extensions that contain roots of a particular polynomial.

Maintained by ynaqvi and last modified 12/17/17