DATE: Spring Semester, 1996

INSTRUCTOR: Professor John Beachy, Northern Illinois University, Tel: 753-6753

OFFICE HOURS: Watson 355, MWF 11:00-11:50, or by appointment.

COURSE OBJECTIVES: The student is expected to continue learning the basic results on groups, rings, and fields. The concepts of homomorphism and quotient structure (using an appropriate equivalence relation) are of particular importance. The course will end with a proof of the impossibility of certain geometric constructions.

COURSE PREREQUISITE:
MATH 420 |
*Recommended*: at least a solid C in 420

TEXT: Beachy/Blair, **Abstract Algebra with a Concrete Introduction**

SYLLABUS: Chapter Three, Groups, (3.6-3.8); Chapter Four, Polynomials (4.1-4.4); Chapter Five, Rings (5.1-5.3); Chapter Six, Fields (6.1-6.3). It will be necessary to cover approximately one section per week.

REFERENCES ON RESERVE:
** A First Course in Abstract Algebra**, by Fraleigh |
** Abstract Algebra**, by Burton |
** Abstract Algebra**, by Herstein

OTHER REFERENCES:
** Contemporary Abstract Algebra**, by Gallian |
** An Introduction to Modern Algebra**, by McCoy |
** Introduction to Abstract Algebra**, by Shapiro

GRADING: Final grades will be based on 600 points: 3 hour tests (300), homework (100), and a comprehensive final exam (200).

WITHDRAWAL: The last day to withdraw from the course without penalty is Friday, March 8.

FINAL EXAM: The final exam is scheduled for Friday, May 10, 10:00-11:50 a.m.

2/9: Test I moved from 2/16 to 2/21; assignment 4 moved fron 2/14 to 2/16

2/19: Assignment 5 moved from 2/23 to 2/26

2/26: Assignment 6 postponed to 3/04, assignment 7 due on 3/08,

3/06: Assignment 8 due 3/20

Su Mo Tu We Th Fr Sa Monday Wednesday Friday 1 2 3 4 5 6 Week of JAN 7 8 9 10 11 12 13 1/15 HOLIDAY 3.6 3.6 14 15 16 17 18 19 20 1/22 3.6 3.7 * 3.7 21 22 23 24 25 26 27 1/29 3.7 3.7 * 3.7 28 29 30 31 1 2 3 2/05 3.8 3.8 * 3.8 FEB 4 5 6 7 8 9 10 2/12 3.8 3.8 4.1 * 11 12 13 14 15 16 17 2/19 4.1 TEST I 4.2 18 19 20 21 22 23 24 2/26 4.2 * 4.3 4.3 25 26 27 28 29 1 2 3/04 4.3 * 4.4 4.4 * MAR 3 4 5 6 7 8 9 3/11 SPRING...................BREAK 10 11 12 13 14 15 16 3/18 4.4 5.1 * TEST II 17 18 19 20 21 22 23 3/25 5.1 5.1 5.2 * 24 25 26 27 28 29 30 4/01 5.2 5.2 5.3 * APR 31 1 2 3 4 5 6 4/08 5.3 5.3 5.3 * 7 8 9 10 11 12 13 4/15 6.1 6.1 TEST III 14 15 16 17 18 19 20 4/22 6.1 6.2 * 6.2 21 22 23 24 25 26 27 4/29 6.2 6.3 * READING DAY 28 29 30 1 2 3 4 5/05 FINAL MAY 5 6 7 8 9 10 11 * denotes a day on which an assignment is due

Due Section Problems to hand in** Recommended problems 1/24 3.6 4 6 7 15 17 1 2 5 11 12 14 1/31 3.7 3 4 5 6 7a,e,f 1 2 8 9 2/07 3.7 11 13 10 12 14 3.8 4 5 12 1 2 3 6 7 8 9 2/16 3.8 13 14 15 20 21 10 11 17 18 22 2/26 4.1 3 4 5 6 7 10 11 13 14 18 3/04 4.2 1b 2a 7b 8b 10 5 9 11 3/08 4.3 2e 2g 3c 3e 4 5 6 3/20 4.4 6 9 12b 12d 13 5 7 8 10 14 3/29 5.1 8 9 10 11 1 2 3 12 15 17 4/05 5.2 4 5 13 15 1 2 3 17 18 4/12 5.3 7 8 9 15 1 2 3 10 11 12 14 4/24 6.1 1d 1f 2 4 1b 1e 3 5 5/01 6.2 1b 1d 1g 5 1c 2 4 5/10 FINAL EXAM Friday, 10-11:50 a.m. **These problems are from the first edition of the book. The numbers below give the corresponding problems in the second edition. Section Problems to hand in Recommended problems 3.6 4 10 11 16 18 1 2 9 5 6 15 3.7 3 4 5 6 7a,d,f 1 2 9 10 3.7 11 12 14 15 13 3.8 4 5 12 1 2 3 6 7 8 9 3.8 13 14 15 20 21 10 11 22 18 24 4.1 9 11 13 14 1 4 5 15 16 20 4.2 1b 4a 2b 5b 10 8 9 11 4.3 2e 3b 4d 5d 6 7 10 4.4 7 12 15b 15d 16 5 9 11 13 17 5.1 8 14 10 11 1 2 3 12 16 19 5.2 4 5 14 15 1 2 3 17 6 5.3 9 19 10 15 1 2 3 11 12 20 14 6.1 1d 1e 3 4 1b 1f 2 5 6.2 1b 1c 2b 6 1d 3 5

Each question is worth 25 points. Each definition is worth 5 points.

**1.**
(a) Define: normal subgroup.

(b) In the dihedral group D_{8} of order 16,
let H={e,a^{2},a^{4},a^{6}}.
Is H a normal subgroup of D_{8}?

(c) In D_{8}, let K={e,a^{4},b,a^{4}b}.
Is K a normal subgroup of D_{8}?

**2.**
Let G = Z_{17}^{x},
the multiplicative group of units of the field Z_{17},
and let N=<4>={1,4,13,16}

(a) Show that N is a subgroup of G.

(b) List the cosets of N in G.

(c) Find the order of each element of G/N.

**3.**
(a) State the division algorithm for polynomials over a field F.

(b) State Eisenstein's irreducibility criterion.

(c) State Kronecker's theorem on the existence of roots.

**4.**
(a) Define: integral domain; prime ideal; maximal ideal.

(b) Give an example of a commutative ring that is not an integral domain.

(c) Give an example of a prime ideal that is not a maximal ideal.

**5.**
(a) Define: minimal polynomial of an element in an extension field.

(b) Show that Q(sqrt(5)+i)=Q(sqrt(5),i).

(c) Find the minimal polynomial of sqrt(5)+i over Q.

**6.**
(a) Define: finite extension field; algebraic element; algebraic extension field.

(b) Prove that any finite extension field is an algebraic extension.

**7.**
Let R and S be commutative rings (with multiplicative identity elements).

(a) Define the addition and multiplication of elements in the
direct sum R(+)S of R and S.

(b) Show that if I is an ideal of R, and J is an ideal of S, then

I(+)J = { (x,y) | x in I and y in J }
is an ideal of R(+)S.

(c) Prove that (R(+)S) / (I(+)J) is isomorphic to R/I (+) S/J.

**8.**
Let G be a group, let N be a normal subgroup of G, and
let H be any subgroup of G.

(a) Assuming that the intersection of H and N is a subgroup,
show that it is normal in H.

(b) Prove that

HN = { x in G | x = yz for some y in H, z in N } is a subgroup of G.

(c)
Prove that
(HN) / N is isomorphic to H / (H intersect N).

Bonus question: Prove that if E is an algebraic extension of K and F is an algebraic extension of E, then F is an algebraic extension of K.

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