# Math 100C - Schedule.

Approximate Lecture Schedule (Beachy and Blair textbook)
It is IMPORTANT to read the material BEFORE the lecture.
 Week ending on Monday Wednesday Friday 1 Apr 6 6.3 6.3 6.3 2 Apr 13 7.1 7.1 7.2 3 Apr 20 7.2 7.3 7.3 4 Apr 27 7.4 7.4 7.4/7.5 5 May 4 7.5 7.5/7.6 Exam 1 6 May 11 7.6 7.7 7.7 7 May 18 8.1 8.1 8.2 8 May 25 8.2 8.3 8.3 9 June 1 Holiday 8.3 Exam 2 10 June 8 8.3 8.4 8.4 11 June 15 8.4 8.4 Final Exam

# Math 100C - Homework Assignments.

•   Homework assignments are due to your TA's box in the basement of AP&M at 5:00pm on Fridays. Late HW will not be accepted. 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. 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 Friday, April 13.
Section 6.3: 1, 2, 3, 4.

HW 2, due on Friday, April 20.
Click on this link to access the HW assignment.

HW 3, due on Friday, April 27.
Clck on this link to access the HW assignement.

HW 4, due on Friday, May 4.
Section 7.4: 2, 7, 8, 11. Section 7.5: 11, 12.

HW 5, due on Friday, May 11.
Section 7.6: 3, 7, 8, 9, 10.
Section 7.7: 2, 4, 6.

HW 6, due on Friday, May 18.
Prove the following "universal properties" in detail.

HW 7, due on Wednesday, May 25.
Section 8.1: 4, 5, 6, 7.
Section 8.2: 5, 7.

HW 8, due on Friday, June 1.
Section 8.3: 1, 3, 5, 7.
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A.
Let R be a PID and let Q(R) be its field of fractions. Let f be a nonconstant polynomial in
R[X]. Prove that f is irreducible in R[X] if and only if f is irreducible in Q(R)[X] and c(f)=R.

Recall that c(f) is the content of f and it is, by definition, the ideal of R generated by the coefficients of f.

Hint. You will need to prove the following generalization of Gauss' lemma: If f, g are in R[X], then c(fg)=c(f)c(g).
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B. Use A to compute the Galois group G(F_q(X^{1/(q-1)})/F_q(X)), where q is a power of
a prime and X is a variable. Is the extension
F_q(X^{1/(q-1)})/F_q(X) Galois ?
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C. Prove that G(Q(X)/Q)  (the Galois group of Q(X)/Q, where X is a variable)
is ismoprphic to the quotient of GL_2(Q) (the group of invertible 2-by-2 matrices
with entries in Q) by its center.

Hint. Use the description of G(Q(X)/Q) given in class and proved during Popescu's office hrs.

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HW 9, due on Friday, June 8.