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Highlighted homework is due soon. |
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Date |
Assignments | ||||
| Apr 24 | pg. 421 Exercises 3, 5, 6, 10, 11 | ||||
| Apr 19 | pg. 409 Exercises 1, 12, 13; pick one of Exercises 5, 6, 11 | ||||
| Apr 17 | pg. 396 Exercises 4,5 | ||||
| Apr 12 | pg. 396 Exercise 1 | ||||
| Apr 10 | pg. 387 Exercises 3, 4, 6 | ||||
| Apr 5 | pg. 376 Exercises 1-3, 7, 13, 14 | ||||
| Apr 3 | pg. 364 Exercises 2, 3, 7-9 | ||||
| Mar 29 | pg. 350 Exercises 1-4, 11 | ||||
| Mar 27 | pg. 335 Exercises 3, 5, 9, 11, 14, 15 | ||||
| Mar 22 | pg. 325 Exercises 1-4, 8 (see Ex. 6.18 for a hint) | ||||
| Mar 20 | pg. 296 Exercises 1, 3, 5, 7, 8 pg. 309 Exercises 1, 2, 8 |
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| Mar 8 | pg. 220 Exercises 1, 2, 8, 25, 29 | ||||
| Mar 6 | pg. 165 Choose 5 of Exercises 6, 9, 12, 14, 18, 19, 23 | ||||
| Mar 1 | This homework is not to be turned in, but you should study it for the final. pg. 129 Exercises 1, 10, 14, 26 Hints: Ex. 10: Use the Root Theorem, which as you should recall is valid in F[x] for any field F. Ex. 14: The Cardano-Tartaglia formula is given in a previous problem. Ex. 26: The back of the book provides an excellent hint. |
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| Feb 22 | This homework is not to be turned in, but you should study it for the test. pg. 114 Warm Ups a, b, d, e, l, m, n, o pg. 116 Exercises 1, 3, 9, 11, 13 Hints: Ex 3: If you don't understand the notation, the problem is asking for two 2×2 matrices A and B. Ex 9: Start with the fact that ab = 1. Multiply both sides by c. It should be straightforward after that. Ex 11(c): Let n be any integer such that n1 = 0. A prime number divides n (say how you know that a prime number divides n). Write this fact as an equation, using the definition of divisibility. Since F is a field, use the associative property and the inverse of an element of F that appears in the equation to show that the prime number is also a multiple of 1 that gives 0. Now, the characteristic is the smallest integer n such that n1 = 0. So, can the characteristic n be composite? If not, why not, and what must the characteristic be? Ex 13: To find the inverse, think about how to factor 1 - x3, 1 - x5, 1 - x7, etc. |
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| Feb 15 | pg. 95 Warm Up c, d, f pg. 96 Exercises 1, 4, 9, 10, 15, 17 Hints: Use Theorem 7.1 whenever possible. WUc. Look at the powers of each element of the set. 15. You will need the Binomial Theorem for part (a). |
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| Feb 13 | pg. 84 Warm Up d, e pg. 85 Exercises 1, 3-5, 12, 13, 15, 18 Hints: 1. Use Theorem 6.1 and the fact that 0 = 0 + 0. 3. Do not try to use -1, since the ring may not contain an identity. You can use Exercise 1 from the text (and probably need to). 4. Apply Exercise 3 from the text. 5. Use the definition of subtraction or Exercise 3 from the text, as appropriate. 12. This problem is mostly a lot of algebra. Don't forget to explain why addition and multiplication are valid operations (i.e., closed). 15. Use Exercise 13 as a guide. In fact I only assigned 13 because you might need a warm up for 15. Otherwise I don't care much for 13 & you shouldn't feel compelled to do it. 18. This problem is awesome once you get it, and maddening until you do. The hint in the text is a little obscure. For part (a), consider the square of x = a + a, where a is an arbitrary element of the ring. Explain how the property of the ring to show that a + a is the same as the square of a + a. Expand the square to a sum of four elements. Use the special property of this ring to simplify each of these elements. Now apply Theorem 6.1. For part (b), repeat with x = a + b, where a and b are arbitrary elements of the ring. |
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| Feb 8 | pg. 64 Warm Up a, b, c, e, g pg. 65 Exercises 1, 4, 5, 7, 8, 13 Hints: 1. Adapt the proof given in class for Theorem 2.8. Use induction on n = deg f. 4. Use factoring by grouping when it is reducible. If it is irreducible, argue why no linear factors exist. See the discussion on pg. 62 of the text. 5. Use the Rational Root Theorem 5.6 to prove that there are no linear factors. Explain why this proves that there are no cubic factors. Theorem 4.1 helps. Thus, there can be only quadratic factors. Writef = pq where p, q are quadratic factors. Write p and q with unknown coefficients and multiply them out. Recall that two polynomials are equal iff corresponding coefficients are equal. What does this say about the coefficients of f and the coefficients of the product pq? You get a system of linear equations whose solutions must be integers. Working with this system leads to a contradiction. 8. The elementary calculus that you should use is the Intermediate Value Theorem. Look it up in a calc text if you don't remember it. If you don't have a calc text handy, pop by the office and I will let you look at mine. In the text currently used by the department, it's Theorem 11 on pg. 130. 13. Use the Rational Root Theorem 5.6. If you use the proof given in class (the "classical" proof), I will give you zero, repeat zero, points. |
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| Feb 6 | pg. 53 Exercise 1 | ||||
| Feb 1 | pg. 39 Warm Up b, c, d, e, f pg. 39 Exercises 3—7, 9 pg. 52 Warm Up a, b, c, d pg. 52 Exercises 3, 5, 6, 13 |
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| Jan 25 | pg. 27 Warm-up e pg. 27 Exercises 6, 7 Show that if a, d, x are integers such that dx divides da, then x divides a. Show that gcd(ad, bd)=d gcd(a, b). pg. 28 Exercise 10 The following problems are not assigned, but you should look at them & remember the results. Let a, b be integers with the unique prime factorizations
pg. 28 Exercises 11-13 |
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| Jan 18 | pg 26 Warm-up a, b, d pg.27 Exercises 1, 2, 4, 8 |
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| Jan 16 |
pg 11 Warm-up c pg 11 Exercises 2—5, 14 Hint for Exercise 3: Rewrite the sum of n numbers as a sum of 2 numbers. For example, 2+3+5=(2+3)+5. |