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| Due date if any | Fuchsia means graded homework. |
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| Section 33 |
Prove the following variation on Theorem 33.8:
For each finite field F there are unique prime p and positive integer n such
that the elements of F are exactly the roots of the polynomial equation x^(p^n)
- x = 0.
Prove Theorem 33.5 by first showing that there is an element α ≠ 0 of some order n such that all other nonzero elements have an order dividing n. Then the powers of this α are all nonzero elements of the finite field. Each such element is called a primitive root of the finite field. |
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| Section 31 | 8, 10, 19, 24, 26 (a little hard, but a short proof) |
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| Section 30 | 2, 6, 7, 15, 24 |
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| Section 29 | 2, 3, 8, 23, 30 and 37 (do 30 and 37 after reading through page 283) |
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| Section 27 | 1, 2, 5, 14, 18, 19, 24 (very tricky; you must show that finite integral domains are fields), 25 |
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| Section 26 | 1, 2, 3, 10bcd, 12, 13, 14, 18, 20 (important but tricky), 30 |
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| Section 23 | 37 (too big for an exam) |
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| 20 March | Find a proof of Theorem 23.11 on page 215 |
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| 10 March | I wish to send my ATM 4-digit code to my friend Pat. I use Pat's public key (n,k) = (14659, 15) to create number r = 5194, for Pat. What is my ATM 4-digit code? |
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| Section 23 | 1, 4, 9, 14, 25, 28, 34, 36 (too big for an exam) |
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| Section 22 | 1, 2, 3, 6, 7, 10, 13, 17, 21, 23, 24 |
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| Section 21 | 2, 4, 12 |
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| Section 20 | 2, 10, 16, 23 |
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| Section 19 | 2, 10, 12 (first compute (a+b)^3), 24, 26 (quite hard, not for exams), 28 |
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| Section 18 | 6, 12, 18, 20, 28 (ignore 27), 38, 46 |
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| 30 January | Prove the 3 group propositions from our website |