- Math 502 Problem Set #8 Due week of Nov. 1, 2010, in lab. Read Artin, Chapter 4, sections 1-2.
- EMBEDDING PROBLEMS AND OPEN SUBGROUPS DAVID HARBATER AND KATHERINE STEVENSON
- Math 502 Problem Set #4 Due week of Oct. 4, 2010, in lab. Read Artin, Chapter 2, sections 6-7.
- Math 603 Problem Set #4 Due Mon., Feb. 21, 2005 1. a) Show that if 0 M M M 0 is an exact sequence of R-modules, then M
- Math 116 Problem Set #8 Due Fri., Nov. 6, 2009, in class. Read Apostol, Chapter 13, sections 18-23, and Chapter 14, sections 1-5.
- Patching and Galois theory David Harbater
- Section 4: Rigid patching This section, like Section 3, discusses an approach to carrying over the ideas of Sec-
- Local Galois theory in dimension two David Harbater
- On function fields with free absolute Galois groups By David Harbater
- Patching over Fields David Harbater
- APPLICATIONS OF PATCHING TO QUADRATIC FORMS AND CENTRAL SIMPLE ALGEBRAS
- THE LOCAL LIFTING PROBLEM FOR ACTIONS OF FINITE GROUPS ON CURVES
- Fields of definition of p-adic covers Pierre D`ebes and David Harbater
- Math 502 Sample Exam #1 October 2010 This exam consists of ten problems. Do them all, showing your work and explaining your
- Math 502 Problem Set #3 Due week of Sept. 27, 2010, in lab. Read Artin, Chapter 2, sections 1-5.
- Math 502 Problem Set #5 Due week of Oct. 11, 2010, in lab. Note: Fall break is Oct. 11-12. If you have a Tuesday lab, you may hand in this assignment
- Math 502 Problem Set #9 Due week of Nov. 8, 2010, in lab. Read Artin, Chapter 4, sections 3-5.
- Math 502 Problem Set #11 Due week of Nov. 22, 2010, in lab. Note: Thanksgiving break is Nov. 25-26. If you have a Thursday lab, you may hand in
- Math 502 Problem Set #12 Due week of Nov. 29, 2010, in lab. Read Artin, Chapter 6, sections 9-11; and Chapter 7, sections 1-3 and 6-7.
- Math 116 Sample Final Exam December, 2009 Instructions: This exam consists of ten problems. Do all ten, showing your work and
- Math 116 Problem Set #2 Due Fri., Sept. 25, 2009, in class. Read Apostol, Chapter 1, sections 9-25, pages 61-83; and Chapter 3, sections 1-2, pages
- Math 116 Problem Set #4 Due Fri., Oct. 9, 2009, in class. Reminder: The first exam is on Monday, October 5 in class. One two-sided handwritten
- Math 116 Problem Set #5 Due Fri., Oct. 16, 2009, in class. Read Apostol, Chapter 4, section 22; and Chapter 12, sections 1-10.
- Math 116 Problem Set #6 Due Fri., Oct. 23, 2009, in class. Read Apostol, Chapter 12, sections 12-14.
- Math 116 Problem Set #7 Due Fri., Oct. 30, 2009, in class. Read Apostol, Chapter 12, section 16; and Chapter 13, sections 1-16.
- Math 116 Problem Set #10 Due Fri., Nov. 20, 2009, in class. Read Apostol, Chapter 8, sections 1-13.
- Math 116 Problem Set #11 Due Mon., Nov. 30, 2009, in class. Read Apostol, Chapter 8, sections 15-27.
- Math 116 Problem Set #12 Due Fri., Dec. 4, 2009, in class. Read Apostol, Chapter 15.
- Math 116 Problem Set #13 December, 2009 This optional problem set may be handed in for credit until noon on Monday, Dec. 14.
- Math 371 Problem Set #2 Due week of Jan. 28, 2008, in lab. Review Herstein, Chapter 1, section 3.
- Math 371 Problem Set #3 Due week of Feb. 4, 2008, in lab. Read Herstein, Chapter 3, sections 1-2.
- Math 371 Problem Set #5 Due week of Feb. 18, 2008, in lab. Reminder: Exam #1 will take place in class on Wednesday, February 20, on the material
- Math 371 Problem Set #6 Due week of Feb. 25, 2008, in lab. Read Herstein, Chapter 3, sections 10-11.
- Math 371 Problem Set #11 Due week of April 7, 2008, in lab. Read Herstein, Chapter 2, sections 7-8.
- Math 370 Exam #2 solutions Nov., 2007 1. We have shown that if a matrix A represents a linear transformation T with respect
- Math 370 Problem Set #6 Due week of Oct. 15, 2007, in lab. Note: Due to Penn's fall break, there will be no class on Monday, Oct. 15 and no lab on
- Math 370 Problem Set #11 Due week of Nov. 19, 2007, in lab. Since there is no Thursday lab this week due to Thanksgiving, those in that lab may submit
- Math 603 Sample Exam Spring 2005 For each of the following, either give an example or explain why none exists.
- Math 603 Problem Set #2 Due Mon., Jan. 31, 2005 1. Suppose that
- Math 603 Problem Set #3 Due Fri., Feb. 11, 2005 1. a) Show that R[x] is a flat R-module.
- Math 603 Problem Set #5 Due Mon., Feb. 28, 2005 1. a) Let V be an affine variety, with ring of functions R. Let W be a Zariski closed
- Math 603 Problem Set #8 Due Mon., April 4, 2005 1. a) Find the degree of =
- Math 603 Problem Set #9 Due Mon., April 11, 2005 1. Let F = Z/pZ, let L = F(x, y), and let K = F(xp
- Math 603 Problem Set #10 Due Mon., April 18, 2005 1. Let p be a prime number.
- Math 602 Sample Exam Fall 2004 For each of the following, either give an example, or else prove that none exists.
- Math 602 Problem Set #3 Due Mon., Oct. 4, 2004 1. a) How many ways can a regular tetrahedron be inscribed in a cube? (Here inscribing is
- Math 602 Problem Set #5 Due Mon., Oct. 18, 2004 1. Let n > 2. Show that if the dihedral group Dn of order 2n is isomorphic to a semi-direct
- Math 602 Problem Set #8 Due Fri., Nov. 12, 2004 1. Let p > 2 be a prime number, and let f(x) = x
- Math 602 Problem Set #9 Due Fri., Nov. 19, 2004 1. Let I1, . . . , In R be ideals in a commutative ring R.
- Math 602 Problem Set #12 Due Mon., Dec. 13, 2004 1. Let V be a vector space and let G End V be a finite subgroup. Say that W V is
- Math 702 Problem Set #4 Due Mon., March 15, 2004 1. Let k be a field with algebraic closure k. Let Y P1
- Math 702 Problem Set #5 Due Fri., March 26, 2004 1. (a) Determine whether or not there is a solution to the equation x2
- Math 702 Problem Set #6 Due Fri., April 9, 2004 1. (a) Let b be an odd positive integer, and let a = 2b. Show that there is a positive
- Math 702 Problem Set #7 Due Mon., April 26, 2004 1. Let Y P1
- Math 625 Problem Set #2 Due Wednesday, Feb. 5, 2003 Read Hartshorne, Chapter IV, section 3.
- Math 625 Problem Set #6 Due Wednesday, April 2, 2003 Read Hartshorne, Chapter III, sections 5-7.
- Math 625 Problem Set #9 Due Wednesday, April 30, 2003 Read Hartshorne, Chapter V, sections 2-6. (Optional: Read Appendix A, sections 3-5;
- Math 624 Problem Set #6 Due Friday, Nov. 8, 2002 Read Fulton, Chapter 4.
- Math 624 Problem Set #8 Due Monday, Nov. 25, 2002 Read Fulton, Chapter 5, sections 4-6.
- Math 624 Problem Set #9 Due Monday, Dec. 2, 2002 Read Fulton, Chapter 6, sections 1-3.
- Math 624 Problem Set #10 Due Monday, Dec. 9, 2002 Read Fulton, Chapter 6, sections 4-6.
- Math 370 Problem Set #10 Due week of Nov. 12, 2007, in lab. Reminder: Exam #2 will take place in class on Monday, November 12, on the material
- Math 370 Problem Set #7 Due week of Oct. 22, 2007, in lab. Read Hoffman and Kunze, Chapter 3, Section 6.
- Math 116 Problem Set #9 Due Fri., Nov.13, 2009, in class. Reminder: The second exam is on Monday, November 9 in class. One two-sided handwrit-
- Math 370 Problem Set #14 Optional problem set. This assignment is optional, and may be handed in at the review session to be held on
- Math 502 Sample Exam #2 December 2010 This exam consists of ten problems. Do them all, showing your work and explaining your
- Math 602 Problem Set #10 Due Mon., Nov. 29, 2004 1. a) If A is a square matrix satisfying A3
- Math 502 Problem Set #10 Due week of Nov. 15, 2010, in lab. Read Artin, Chapter 4, sections 6-7.
- Math 625 Problem Set #4 Due Wednesday, March 5, 2003 Read Hartshorne, Chapter IV, section 5.
- Math 370 Problem Set #12 Due week of Nov. 26, 2007, in lab. Review Hoffman and Kunze, Chapter 6, Sections 1-2.
- Math 704 Problem Set #3 Due Mon., Feb. 23, 2004 1. Let C be a curve of genus 2 over a field k (which for simplicity may be assumed to
- Math 370 Problem Set #9 Due week of Nov. 5, 2007, in lab. Read Hoffman and Kunze, Chapter 4, Section 4.
- Math 371 Problem Set #14 Due week of April 28, 2008, in lab. Note: There is no lab on Thursday, May 1; the last lab is on Tuesday, April 29, and all
- Math 370 Problem Set #13 Due week of Dec. 3, 2007, in lab. Read Hoffman and Kunze, Chapter 6, Section 3, and Chapter 8, Sections 1 and 2.
- Math 371 Problem Set #7 Due week of March 3, 2008, in lab. Read Herstein, Chapter 3, section 8, and Chapter 5, section 1.
- FUNDAMENTAL GROUPS AND EMBEDDING PROBLEMS IN CHARACTERISTIC p
- Math 602 Problem Set #7 Due Fri., Nov. 5, 2004 1. Prove, or disprove and salvage: If K is a field, and f(x) K[x] has no roots, then
- Math 603 Problem Set #11 Due Wed., April 27, 2005 1. Suppose k K is a separable field extension of degree n.
- Math 603 Problem Set #6 Due Mon., March 14, 2005 1. a) Let R be a Noetherian ring, I the set of ideals of R, and I0 a subset of I. Let
- Patching and Galois theory David Harbater
- Math 624 Problem Set #7 Due Friday, Nov. 15, 2002 Read Fulton, Chapter 5, sections 1-3.
- Correction and Addendum to "Embedding Problems with Local Conditions" David Harbater
- Math 625 Problem Set #3 Due Wednesday, Feb. 19, 2003 Read Hartshorne, Chapter IV, section 4.
- GALOIS GROUPS WITH PRESCRIBED RAMIFICATION David Harbater
- Math 602 Problem Set #4 Due Mon., Oct. 11, 2004 1. Let G be a p-group, and let be its Frattini subgroup.
- Math 371 Problem Set #1 Due week of Jan. 21, 2008, in lab. Read Herstein, Chapter 1.
- Math 116 Sample Exam #1 October, 2009. Instructions: This exam consists of five problems. Do all five, showing your work and
- Math 702 Problem Set #1 Due Wed., Jan. 28, 2004 1. Let R be a Noetherian domain, and let p be a non-zero prime ideal of R. Let Rp be the
- Math 602 Problem Set #2 Due Mon., Sept. 27, 2004 1. Let G be the symmetry group of a regular polyhedron P, let v be a vertex of P, and consider
- Math 625 Problem Set #5 Due Friday, March 21, 2003 Read Fulton, Chapter 8, section 6.
- Math 350 Sample Final Exam December 2005 1. Show that there are infinitely many primes of the form 6k -1, with k a positive
- Math 704 Problem Set #2 Due Mon., Feb. 9, 2004 1. Consider the extension Z Z[], where = 1+
- Math 603 Problem Set #1 Due Mon., Jan. 24, 2005 1. Which of the following R-modules are finitely generated? Which are free? Which are
- PATCHING SUBFIELDS OF DIVISION ALGEBRAS DAVID HARBATER, JULIA HARTMANN, AND DANIEL KRASHEN
- Math 603 Problem Set #7 Due Fri., March 25, 2005 1. Which of the following rings R are discrete valuation rings? For those that are, find
- Math 116 Sample Exam #2 November, 2009 Instructions: This exam consists of five problems. Do all five, showing your work and
- Local Galois theory in dimension two: Second edition David Harbater
- Fundamental Groups of Curves in Characteristic p David Harbater*
- Math 502 Problem Set #1 Due week of Sept. 13, 2010, in lab. Read Artin, Chapter 1, sections 1-3.
- Math 502 Problem Set #7 Due week of Oct. 25, 2010, in lab. Reminder: The first exam will take place in class on Wednesday, October 27. It will cover
- Math 624 Problem Set #4 Due Wednesday, Oct. 23, 2002 Read Fulton, Chapter 2, sections 5-11, and Chapter 3, section 1.
- Noname manuscript No. (will be inserted by the editor)
- Math 370 Problem Set #5 Due week of Oct. 8, 2007, in lab. Reminder: Exam #1 will take place in class on Monday, October 8, on the material covered
- Math 602 Problem Set #11 Due Mon., Dec. 6, 2004 1. Let V, W, Y be finite dimensional vector spaces over K.
- Fields Institute Communications Volume 00, 0000
- Approximating Galois orbits of dessins David Harbater* and Leila Schneps**
- Math 621 Problem Set #3 Due Wed., Feb. 20, 20* 1. Let A be a complete discrete valuation ring with fraction field K, let L be *
- Math 620 Problem Set #8 Due Mon., Nov. 5, 2001 1. a) Show directly that there are infinitely many primes -1(mod 6).
- Fundamental groups of moduli and the Grothendieck-Teichm"uller group David Harbater* and Leila Schneps
- 1 Embedding Problems and Adding Branch Points
- Math 625 Problem Set #8 Due Monday, April 21, 2003 Read Hartshorne, Chapter III, sections 11-12; Chapter V, section 1; Appendix A, sections
- Math 602 Problem Set #1 Due Mon., Sept. 20, 2004 1. Define the center of a group G to be Z = {g G | (h G) gh = hg}.
- Math 625 Problem Set #1 Due Wednesday, Jan. 29, 2003 Read Hartshorne, Chapter IV, sections 1,2.
- Math 602 Problem Set #6 Due Fri., Oct. 29, 2004 1. Which of the following are rings? (Note: To be a ring, it must have a multiplicative
- Math 620 Problem Set #9 Due Wed., Nov. 14, 2001 1. For d = 2, 3, 5, 11, 13
- Math 621 Problem Set #2 Due Mon., Feb. 11, 2002 1. Let A be a complete discrete valuation ring with respect to an absolute valu*
- Math 620 Problem Set #7 Due Mon., Oct. 29, 2001 1. a) For which primes p is there a solution to x2 = 13 in Z=p? in Zp? in Qp?
- Abhyankar's Local Conjecture on Fundamental Groups David Harbater* and Katherine F. Stevenson
- Math 620 Problem Set #10 Due Mon., Nov. 26, 2001 1. If a, b 2 Z are relatively prime and b > 0, let I(a=b) be the open interval*
- Shafarevich Conjecture -The Shafarevich Conjecture in inverse Galois theory asserts that the absolute Galois group GQab := Gal(Q~=Qab ) of Qab is a free pr*
- Math 621 Problem Set #1 Due Wed., Jan. 30, 2002 1. Let M be the set of p-adic absolute values on Q (including the "usual" absol*
- Math 502 Problem Set #13 Due the week of Dec. 6, 2010, in Tues. lab or in class. Reminders: There will be an exam in class on Wed., Dec. 8. It will focus on material
- Generalizations of Abhyankar's Conjecture An appendix to Desingularization and modular Galois theory by S.S. Abhyankar
- Math 620 Problem Set #4 Due Fri., Oct. 5, 2001 1. a) Show that the ring of p-adic integers Zp is a local ring with maximal id*
- Math 620 Problem Set #6 Due Mon., Oct. 22, 2001 1. A function f on N is called multiplicative if f(ab) = f(a)f(b) provided (a, *
- Math 620 Problem Set #1 Due Fri., Sept. 14, 2001 1. a) Use the Euclidean algorithm to find the g.c.d. of 110 and 39, and to fi*
- Math 620 Problem Set #3 Due Fri., Sept. 28, 2001 P n
- Abhyankar's Conjecture and embedding problems David Harbater*
- Math 620 Problem Set #2 Due Fri., Sept. 21, 2001 1. a) Show that there is a uniqueXinteger-valued function ~ : N ! Z on the na*
- Math 621 Problem Set #5 Due Wed., March 20, 2002 1. a) Redo ProblemQSet 4, problem 3, with i5 replaced by im , for an arbitrary*
- Math 621 Problem Set #6 Due Fri., April 5, 2002 1. Let K be a number field, and let f(x) 2 K[x] be a polynomial with distinct r*
- Embedding Problems with Local Conditions David Harbater*
- Math 621 Problem Set #4 Due Mon., March 4, 20* p _ p _
- Math 621 Problem Set #7 Due Fri., April 19, 2002 p__
- Math 620 Problem Set #11 Due Fri., Dec. 7, 2001 1. Let p be a prime, and let a, b be integers such that a2 + b2 + 1 0 (mod p*
- Patching and Thickening Problems David Harbater* and Katherine F. Stevensony
- Math 620 Problem Set #5 Due Mon., Oct. 15, 2001 1. Let K be a field, with multiplicative group K*. Let G be a finite subgroup o*
- Patching and Thickening Problems David Harbater
- Local-global principles for torsors over arithmetic curves David Harbater, Julia Hartmann, and Daniel Krashen
- Math 503 Problem Set #3 Due the week of Jan. 29, 2007, in lab. Read Artin, Chapter 10, sections 5-8.
- Quadratic Forms (Math 520/620/702) Fall 2011 Suggested topics for presentations
- Math 503 Problem Set #11 Due the week of April 2, 2007, in lab. Read Artin, Chapter 13, sections 3-5.
- Math 503 Problem Set #6 Due the week of Feb. 19, 2007, in lab. Read Artin, Chapter 11, sections 4-11.
- Math 503 Problem Set #4 Due the week of Feb. 5, 2007, in lab. Read Artin, Chapter 10, sections 7,8; Chapter 11, section 1,2.
- Math 503 Problem Set #10 Due the week of March 26, 2007, in lab. Read Artin, Chapter 13, sections 1-3.
- Math 503 Problem Set #1 Due the week of Jan. 15, 2007, in lab. Read Artin, Chapter 10, sections 1-3 and 6.
- Math 502 Problem Set #7 Due the week of Oct. 23, 2006, in lab or in class. Reminders: There will be an exam in class on Wed., Oct. 25, on the material covered up
- Math 503 Problem Set #13 Due the week of April 16, 2007, in lab. Reminder: There will be an exam in class on Wed., April 18, on the material beyond what
- Math 502 Problem Set #10 Due the week of Nov. 13, 2006, in lab. Read Artin, Chapter 4, sections 5 and 7.
- Quadratic Forms (Math 520/620/702) Problem Set #4 Due Wed., Nov. 23, 2011, in class.
- Quadratic Forms (Math 520/620/702) Problem Set #2 Due Mon., Oct. 24, 2011, in class.
- Quadratic Forms (Math 520/620/702) Problem Set #3 Due Wed., Nov. 9, 2011, in class.
- Math 503 Problem Set #2 Due the week of Jan. 22, 2007, in lab. Read Artin, Chapter 10, sections 4-7.
- Math 503 Problem Set #9 Due the week of March 19, 2007, in lab. Read Artin, Chapter 12, sections 5-7.
- Math 502 Problem Set #5 Due the week of Oct. 9, 2006, in lab. Read Artin, Chapter 2, sections 7-10.
- Math 503 Problem Set #12 Due the week of April 9, 2007, in lab. Read Artin, Chapter 13, sections 6-9; and Chapter 14, section 1.
- Math 503 Problem Set #5 Due the week of Feb. 12, 2007, in lab. Read Artin, Chapter 11, sections 1-3, 5.
- Math 502 Problem Set #11 Due the week of Nov. 20, 2006, in lab or class. Note: There is no lab this week on Thursday, due to Thanksgiving. Those in the Thursday
- Math 502 Problem Set #6 Due the week of Oct. 16, 2006, in lab. Read Artin, Chapter 3, sections 1-5.
- Math 502 Problem Set #8 Due the week of Oct. 30, 2006, in lab. Review Artin, Chapter 4, sections 1-3.
- Math 502 Problem Set #9 Due the week of Nov. 6, 2006, in lab. Read Artin, Chapter 4, sections 4 and 6.
- Quadratic Forms (Math 520/620/702) Problem Set #1 Due Wed., Oct. 5, 2011, in class.
- Weierstrass preparation and algebraic invariants David Harbater, Julia Hartmann, and Daniel Krashen
- Quadratic Forms (Math 520/620/702) Problem Set #5 Due Wed., Dec. 7, 2011, in class.
- Math 502 Problem Set #13 Due the week of Dec. 4, 2006, in Tues. lab or in class. Reminders: There will be an exam in class on Wed., Dec. 6, focusing on the material
- Math 503 Problem Set #7 Due the week of Feb. 26, 2007, in lab. Reminder: There will be an exam in class on Mon., Feb. 26, on the material covered up