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

The final exam will be given out on Friday, May 6, at the end of class. You will have until 5:00pm on Thursday, May 12, to complete this take-home exam.

Reading/exam period office hours are Monday 4:00pm-6:00pm, Tuesday 2:00-4:00pm.

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.

For this exam, you may use the course textbook, your class notes, your old homework sets and exam, and the solutions to these which were posted on the course Moodle page. You may not use any other outside help, including other websites or texts, and you may not discuss the exam or related mathematics with anyone else.

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 homomorphisms 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. Be familiar with 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 and exercises so that you are able to construct field extensions that contain roots of a particular polynomial.

21: Know the definition of terms divides, prime, irreducible, unit, associates, PID, UFD and Euclidean domain. You should also know that every Euclidean domain is a PID and that every PID is a UFD, and you should be able to use this to show that the ring of integers and F[X] for some field F are both UFDs. You should also know the statements and proofs of Theorems 21.1 and 21.4. Know how to use the Euclidean Algorithm to obtain the gcd and the x and y mentioned in Exercise 21.11. Be familiar with the domains given in this section that are not UFDs, both from the examples and the exercises.