a multiplicative subgroup, math homework help

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Math 5420 Homework 10 Due July 25, 2017 1. If G = R× and N = {−1, 1}, show how you can ‘identify’ G/N as the group of all positive real numbers under multiplication. What are the cosets of N in G? 2. p. 176 # 18 3. p. 176 # 21 4. p. 176 # 23 5. p. 177 # 27 6. p. 190 # 8 7. p. 190 # 13 8. Find the quotient and the remainder when x3 + 2 is divided by 2x2 + 3x + 4 in Z3 and Z5 . 9. p. 200 # 1 a, d 10. p. 201 #2 b, c 11. p. 201 #5 a, d over 5) Find the greatest common divisor of the given polynomials, over the given field. +(a) x4 + x3 + x + 1 and x3 + x² + x + 1 Z2 (b) x3 – 2x2 + 3x + 1 and x3 + 2x + 1 Z5 +(c) x5 + 4x4 + 6x3 + 6x2 + 5x + 2 and x + 3x² + 3x + 6 27 (d) x5 + x4 + 2x2 + 4x + 4 and x3 + x² + 4x 25 over over over
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1. Obviously 𝑁 is a (multiplicative) subgroup in ℝ× (all real numbers with multiplication).
The group is abelian, thus all its subgroups are normal and factorization is possible.
The cosets of 𝑁 are the sets 𝑥𝑁 = {𝑥𝑦: 𝑦 ∈ 𝑁} = {𝑥𝑦: 𝑦 = ±1} = {𝑥, −𝑥} (sets with two
elements each, one is positive and one is negative).
×
To identify 𝑁 with ℝ�...


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