# Math 100B - Schedule.

Approximate Lecture Schedule (Beachy and Blair text)
It is IMPORTANT to read the material BEFORE the lecture.
 Week ending on Monday Wednesday Friday 1 Jan 7 2 Jan 14 3 Jan 21 Holiday 4 Jan 28 Review Exam 1 5 Feb 4 6 Feb 11 7 Feb 18 Review 8 Feb 25 Holiday Exam 2 9 Mar 4 10 Mar 11 11 Mar 18 Final Exam

FINAL EXAM: Monday, March 14, 11:30am - 2:30pm.

# Math 100B - Homework Assignments.

• Please drop off your HW in the HW box assigned to this course and located on the 6th floor of the AP&M building (to the right as you exit the elevators) every Wednesday by 5:00pm. The homework assignments have to be written up neatly on letter size paper. The pages have to be stapled together. Students are allowed to discuss the homework among themselves, but are expected to turn in their own work — copying someone else's is not acceptable. No late HW will be accepted. Homework scores will contribute 20% to the final grade.
• Although you are required to turn in only the HW problems listed below, you are strongly advised to attempt solving as many problems from each section as possible.

HW 1, due on Wednesday, January 12.
Section 5.1: 2d, 2e, 4, 7, 11, 15, 16, 19.

HW 2, due on Wednesday, January 19.
Section 5.2: 1, 2, 3, 6, 11, 16, 19.

HW 3, due on Wednesday, January 26.
Section 5.2: 8, 13, 14, 15, 18, 22, 23.

HW 4, due on Wednesday, February 2.
Section 5.3: 11, 12, 13, 14, 16.

HW 5, due on Wednesday, February 9.
Section 5.3: 15, 17, 18, 19, 20, 21, 22, 26.

HW 6, due on Wednesday, February 16.
Section 5.4: 4, 6, 7, 8, 9, 10, 11, 12.

HW 7, due on Wednesday, February 23.

HW 8, due on  Wednesday, March 2.
Section 4.3: 6, 7, 9, 10, 14, 15.

Additional problem: Assume that R is a commutative ring and I is an ideal in R.
(1) Prove that I[X] is an ideal in R[X]. (Note: I[X] denotes the set of polynomials in R[X] whose coefficients are in I.)
(2) Show that the rings R[X]/I[X] and R/I[X] are isomorphic.
(2) Show that if I is a prime ideal in R, then I[X] is a prime ideal in R[X].
(3) Is it true that if I is a maximal ideal in R, then I[X] is a maximal ideal in R[X]? Justify your answer.

HW 9, due on Wednesday, March 9.

1. Use universality properties to show that if R is an integral
domain, then the rings Q(R[X]) and Q(R)(X) are isomorphic.

2. Let R be a commutative ring.
a) Prove that the set Nil(R) consisting of all the nilpotent elements
in R is equal to the intersection of all the prime ideals in R.
In particular, observe that Nil(R) is an ideal in R. It is called
the nilradical of R.
b) Prove that if R is a field, then the units in R[X] coincide with those
in R.
c) Prove that a polynomial f in R[X] is a unit in R[X] if and only if
its constant coefficient is a unit in R and its other coefficients
are nilpotent elements in R. (Hint: Use the reduction morphisms
R[X] --> R/P[X], for all prime ideals P in R.)

3. Let R be an integral domain. A polynomial f in R[X] is called irreducible if it
cannot be written as a product f=gh, with g and h non units in R[X].
a) Prove that if R is a field, then any polynomial in R can be written as
a product of irreducible polynomials.
b) Prove that if R is a field, then an non-zero ideal I in R[X] is prime if and only if
it is generated by an irreducible polynomial.
c) Prove that if R is a field, then any non-zero prime ideal in R[X] is maximal.

HW 10 - will not be collected.