# Math 100C - Schedule.

Approximate Lecture Schedule (Beachy and Blair textbook)
It is IMPORTANT to read the material BEFORE the lecture.
 Week ending on Tuesday Wednesday Thursday 1 Apr 4 Overview 7.3 2 Apr 11 7.4 7.4 3 Apr 18 7.5 7.5 4 Apr 25 7.6 7.6, 7.7 5 May 2 Exam 1 7.7, 8.1. 6 May 9 8.1 8.1 7 May 16 8.2 8.2 8 May 23 8.3 8.3 9 May 30 Exam 2 8.3 8.3+8.4 10 June 6 11 June 13 Final Exam

# Math 100C - Homework Assignments.

•   Homework assignments are due to your TA's box in the basement of AP&M at 6:00pm 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 11.
Read Chapter 7 (review 7.1-7.2; read 7.3 by the end of week 1; read 7.4 - 7.5 be the end of week 2.)
Submit solutions for the following problems: Section 7.3: 2, 3, 6, 7, 12, 13.

HW 2, due on Friday, April 18.
Read: Example 7.4.2, Propositions 7.4.4 and 7.4.6, Lemma 7.5.9, Theorem 7.5.10, Lemma 7.5.11, Theorem 7.5.12, Corollary 7.5.13.
Submit solutions for the following problems: Section 7.4: 2, 4, 5, 8, 11, 12. Section 7.5: 4.

HW 3, due on Friday, April 25.
Read by Tuesday: proof of Lemma 7.5.4.
Submit solutions to the following problems from Section 7.5: 3, 5, 6, 7, 8, 9, 11, 12.

HW 4, due on Monday, May 5 (by 6:00pm.)
Read (by Thursday, May 1): Definition 7.6.9, Theorem 7.6.10, Theorem 7.7.4.
Submit solutions to the following: Section 7.6: 3, 7, 8, 9, 10. Section 7.7: 2, 3, 4, 6.

HW 5, due on Tuesday, May 13 (by 6:00pm.)
Section 8.1: 1,2,3,4,5,6,7.

HW 6, due on Wednesday, May 21 (by 6:00pm.)
Section 8.2: 1,2,3,4,5,7,8.

Extra credit problem I (no partial credit, no help, due on final exam day; 20 pts.)
Let G be a finite group. Assume that  there exist a prime number p and an element a in G different from e
such that axa^{-1}=x^{p+1}, for all x in G. Show that:
1) G is a p-group.
2) H:= { x \in G | x^p=e } is a subgroup of G.
3) |H|^2 > |G|.

HW 7, due on Monday, June 2 (by 6:00pm).
Section 8.3: 1,2,3,4,5,6,7.

Extra credit problem 2. (10 points)

Let X, Y be two (algebraically) independent variables over the finite
field F_p. Prove that the  field extension
F_p(X^{1/p}, Y^{1/p})/F_p(X, Y)
is not simple.

Extra credit problem 3 (15 points.)
Let A be a non-trivial commutative ring, such that
for all nonzero x\in A there exist natural numbers m, n, such that
(1+x^m)^n=x.
1) Prove that A is a field.
2) Prove that any endomorphism of A (ring morphism f : A --> A)
is an automorphism (is bijective.)

Extra credit problem 4 (10 points.)
Give an example of a ring A satisfying the properties in problem 3 above.

HW 8 (will not be collected, but it is part of the final exam material.)
Section 8.4: 1,2,3,4,5,7,8.

----------------------------------------
Let E/F a finite field extension and K an algebraically closed
field containing F as a subfield. Let \sigma: F-->K be a field
morphism.
1) Prove that the number of distinct field morphisms
\tau: E--->K which extend \sigma is less than or equal
to [E:F].
2) Prove that you have equality in 1) above if and only if
the extension E/F is separable.
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The problem above should help you give a very quick and elegant
solution to problem 7, section 8.3.