| 29 April |
Section 22.4 :
If you were to factorize polynomial x^(3^6) - x = x^729 - x over Z_3 into
monic prime factors, how many prime factors would there be of degree 1, of degree 2, etc? If you were to factorize polynomial x^(3^6) - x = x^729 - x over GF(3^2) into monic prime factors, how many prime factors would there be of degree 1, of degree 2, etc? |
| 26 April | Section 22.4 (page 308): 17, 24 |
| 24 April |
Section 22.4: Factorize X^4 - X into primes over GF(4), where GF(4) has an
element \alpha such that \alpha^2 + \alpha + 1 = 0. Factorize X^8 - X into primes over GF(4), where GF(4) has an element \alpha such that \alpha^2 + \alpha + 1 = 0. Factorize X^16 - X into primes over GF(4), where GF(4) has an element \alpha such that \alpha^2 + \alpha + 1 = 0. |
| 22 April | Section 22.4 (page 308): 20 (needs counting), 21, 22, 23 |
| 19 April |
Section 22.4 (page 308): 14, 15 (tricky, a simple proof when the size is
even, and a pigeon hole principle when the size is odd), 16, 18 Homeworks for points are in this color Factorize polynomial x^(2^6) - x = x^64 - x over Z_2 into monic prime factors. How many prime factors are here of degree 1, of degree 2, etc? If you were to factorize polynomial x^(2^15) - x = x^32768 - x over Z_2 into monic prime factors, how many prime factors would there be of degree 1, of degree 2, etc? If you were to factorize polynomial x^(3^8) - x = x^6561 - x over Z_3 into monic prime factors, how many prime factors would there be of degree 1, of degree 2, etc? Due date Monday 29 April. |
| 17 April | Section 22.4 (page 308): 3, 7b |
| 15 April | Section 22.4 (page 308): 7bcd, 8, 12 |
| 12 April | Section 22.4 (page 308): 5, 6, 7a |
| 10 April | Section 22.4 (page 308): 1ab, 2, 4 |
| 8 April |
Section 21.5 (page 293): 11, 12 (not), 13 Find the splitting field of p(x) = x^4 - 2 over the field of rationals Q. |
| 5 April |
Section 21.5 (page 293): 5, 9, 10 Find the dimension [E:F] where F equals the rationals Q and E equals Q(\sqrt{1 + \sqrt{2}}), and give a basis of the vector space E over F. |
| 3 April | Section 21.5 (page 293): 3cd, 4a |
| 25 March | Section 21.5 (page 293): 2dgh, 3ab |
| 22 March | Section 21.5 (page 293): 1cde, 2ab |
| 20 March |
Section 21.5 (page 293): 1ab Section 20.5 (page 272): 16, 17 |
| 18 March | Section 20.5 (page 272): 4, 5, 11, 15 |
| 8 March | Section 20.5 (page 272): 3, 7 |
| 6 March | Section 20.5 (page 272): 1, 2, 4, 5 |
| 4 March |
Section 18.4 (page 248): 16 (factorize 6), 17 (no) Homeworks for points are in this color Factor integers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13 into irreducibles over the Gaussian integers Z[i]. Show that 2 + \sqrt{2} divides 2 + 3 \sqrt{2} in Z[\sqrt{2}] by computing the quotient in Z[\sqrt{2}]. Factor integers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13 into irreducibles over the Euclidean integral domain Z[i \sqrt{2}]. Due date Wednesday 20 March (after Spring break). |
| 1 March | Section 18.4 (page 248): 1, 2, 18 (tricky, but actually easy) |
| 28 February | Section 18.4 (page 248): 11d, 14, 15 |
| 26 February | Section 18.4 (page 248): 11b (very difficult), 11c, 12, 13 |
| 23 February | Section 18.4 (page 248): 16, 17 |
| 21 February | Section 18.4 (page 248): 10, 11a, 13 |
| 19 February | Section 18.4 (page 248): 4, 5, 9 |
| 14 February | Section 18.4 (page 248): 1, 2 |
| 12 February | Section 17.5 (page 231): 25bce, 26, 27 |
| 9 February | Section 17.5 (page 231): 18, 20, 21 (easy if F is infinite), 24 |
| 7 February | Section 17.5 (page 231): 12, 14 (tricky), 15 (tricky) |
| 5 February | Section 17.5 (page 231): 6, 7, 9, 10 |
| 2 February | Section 17.5 (page 231): 4a, 5bd |
| 31 January |
Section 17.5 (page 231): 2cf, 3c Section 16.7 (page 213): 36, 37 |
| 29 January |
Section 16.7 (page 213): 29, 30 (disprove), 33 Homeworks for points are in this color For which positive integers n has the finite ring Z_n non-zero nilpotent elements, and why? In such cases, describe the ideal of nilpotent elements. Due date Friday 9 February. |
| 26 January | Section 16.7 (page 213): 27, 28 |
| 24 January | Section 16.7 (page 213): 25, 26 |
| 22 January | Section 16.7 (page 213): 7, 8, 11, 13a, 16, 24 |
| 19 January |
Section 16.7 (page 213): 4, 5, 6 An axiomatization of groups is at 180923_aata_grp.pdf |
| 17 January | Section 16.7 (page 213): 1, 2, 3 |