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9. In Z/31Z, using Proposition 3, find a primitive root modulo 31. Proposition 3. Let a,b...
4. Recall that a € ZnZ is called a primitive root modulo n, if the order...
4. Recall that a € ZnZ is called a primitive root modulo n, if the order of a in Z/nZ is equal to on). We have seen in class that, if p is a prime, then we can always find primitive roots modulo p. Find all elements of (Z/11Z)* that are primitive roots modulo 11. 5. Can you find primitive roots modulo 16? Explain your answer. 6. In class, we found 2 primitive roots modulo 9 = 32, namely 2...
Let p be an odd prime. Prove that if g is a primitive root modulo p, then g^(p-1)/2 ≡ -1 (mod p).
Let
p be an odd prime. Prove that if g is a primitive root modulo p,
then g^(p-1)/2 ≡ -1 (mod p).
Let p be an odd prime. Prove that if g is a primitive root modulo p, then go-1)/2 =-1 (mod p) Hint: Use Lemma 2 from Chapter 28 (If p is prime and d(p 1), then cd-1 Ξ 0 (mod p) has exactly d solutions).
Let p be an odd prime. Prove that if g is a primitive...
Let n = rs, where r and s are distinct odd primes. Show that there is...
Let n = rs, where r and s are distinct odd primes. Show that there is not a primitive root modulo n. (Solve through letting L = LCM(r − 1, s − 1). Show that the order of any element of (Z/nZ)∗ is at most L, and that L < φ(n).)
g 2 is a primitive root modulo 19. Use the following table to assist you in...
g 2 is a primitive root modulo 19. Use the following table to assist you in the solution of the first two questions and 4(a). The most efficient solutions involve the use of the table and the application of theory; numerically correct solutions involving long computations will not receive full credit t 1234567 8 9 10 11 12 13 14 15 16 1718 2 48 16 13 7 149 18 1715 11 3 6 12 5 10 1 (a) Find...
2. Let G be an abelian group. Suppose that a and b are elements of G...
2. Let G be an abelian group. Suppose that a and b are elements of G of finite order and that the order of a is relatively prime to the order of b. Prove that <a>n<b>= <1> and <a, b> = <ab> .
Problem 2. Find a primitive root for 53. Using this, you can devise a bijection α...
Problem 2. Find a primitive root for 53. Using this, you can
devise a bijection α from the integers modulo 52 to the nonzero
integers modulo 53 with the property that α(a + b) = α(a)· α(b)
modulo 53. Explain. Does the law of exponents get involved at all?
Note: For this to work right, you can think of integers mod 52 as
{0, 1, 2, . . . , 51} or as any complete system of residues modulo
52,...
G-2 is a primitive root modulo 19. Use the following table to assist you in the solution of the f...
g-2 is a primitive root modulo 19. Use the following table to assist you in the solution of the first two questions and 4(a). The most efficient solutions involve the use of the table and the application of theory; numerically correct solutions involving long computations will not receive full credit t1 2 3 4 567 89 10 11 12 13 14 151617 18 g 2 481613714918 17 15 11 36125101 Question 1. (a) Find the least positive residue of 126...
(b) Let p be a prime that is congruent to 3 modulo 4. Let b ∈...
(b) Let p be a prime that is congruent to 3 modulo 4. Let b ∈ Z. Let a = b (p+1)/4 . Show that a 2 ≡ ±b (mod p). (c) Give an algorithm to compute square roots of something modulo p, when p ≡ 3 (mod 4). Note: Not all things are square modulo p, so the algorithm should return the square root or inform you there is no square.
(a) Let p be a prime number, let H be a subgroup of (Z/pZ), and let a identify (Z/pZ) Question 9. EieH . Prove that...
(a) Let p be a prime number, let H be a subgroup of (Z/pZ), and let a identify (Z/pZ) Question 9. EieH . Prove that Q(a) equals the fixed field Q(,)". (Here with the Galois group in the usual way.) (b) Draw the diagram of all subfields of Q(13) and find primitive elements for each of them. we
(a) Let p be a prime number, let H be a subgroup of (Z/pZ), and let a identify (Z/pZ) Question 9. EieH...
6. (16 points) Let CE C be a primitive n-th root of unity. Let X =...
6. (16 points) Let CE C be a primitive n-th root of unity. Let X = 6 + 1/5. (a) (4 points) Show that Q(5) R = Q(1). (b) (4 points) Let f be the minimal polynomial of over Q. Show that Q(x) is a splitting field of f over Q. (c) (4 points) Show that Gal(Q(^)/Q) – (Z/nZ)* / (-1). (d) (4 points) Find the minimal polynomial of 2 cos(27/9) over Q.
4. Recall that a € ZnZ is called a primitive root modulo n, if the order of a in Z/nZ is equal to on). We have seen in class that, if p is a prime, then we can always find primitive roots modulo p. Find all elements of (Z/11Z)* that are primitive roots modulo 11. 5. Can you find primitive roots modulo 16? Explain your answer. 6. In class, we found 2 primitive roots modulo 9 = 32, namely 2...
Let
p be an odd prime. Prove that if g is a primitive root modulo p,
then g^(p-1)/2 ≡ -1 (mod p).
Let p be an odd prime. Prove that if g is a primitive root modulo p, then go-1)/2 =-1 (mod p) Hint: Use Lemma 2 from Chapter 28 (If p is prime and d(p 1), then cd-1 Ξ 0 (mod p) has exactly d solutions).
Let p be an odd prime. Prove that if g is a primitive...
g 2 is a primitive root modulo 19. Use the following table to assist you in the solution of the first two questions and 4(a). The most efficient solutions involve the use of the table and the application of theory; numerically correct solutions involving long computations will not receive full credit t 1234567 8 9 10 11 12 13 14 15 16 1718 2 48 16 13 7 149 18 1715 11 3 6 12 5 10 1 (a) Find...
2. Let G be an abelian group. Suppose that a and b are elements of G of finite order and that the order of a is relatively prime to the order of b. Prove that <a>n<b>= <1> and <a, b> = <ab> .
Problem 2. Find a primitive root for 53. Using this, you can
devise a bijection α from the integers modulo 52 to the nonzero
integers modulo 53 with the property that α(a + b) = α(a)· α(b)
modulo 53. Explain. Does the law of exponents get involved at all?
Note: For this to work right, you can think of integers mod 52 as
{0, 1, 2, . . . , 51} or as any complete system of residues modulo
52,...
g-2 is a primitive root modulo 19. Use the following table to assist you in the solution of the first two questions and 4(a). The most efficient solutions involve the use of the table and the application of theory; numerically correct solutions involving long computations will not receive full credit t1 2 3 4 567 89 10 11 12 13 14 151617 18 g 2 481613714918 17 15 11 36125101 Question 1. (a) Find the least positive residue of 126...
(a) Let p be a prime number, let H be a subgroup of (Z/pZ), and let a identify (Z/pZ) Question 9. EieH . Prove that Q(a) equals the fixed field Q(,)". (Here with the Galois group in the usual way.) (b) Draw the diagram of all subfields of Q(13) and find primitive elements for each of them. we
(a) Let p be a prime number, let H be a subgroup of (Z/pZ), and let a identify (Z/pZ) Question 9. EieH...
6. (16 points) Let CE C be a primitive n-th root of unity. Let X = 6 + 1/5. (a) (4 points) Show that Q(5) R = Q(1). (b) (4 points) Let f be the minimal polynomial of over Q. Show that Q(x) is a splitting field of f over Q. (c) (4 points) Show that Gal(Q(^)/Q) – (Z/nZ)* / (-1). (d) (4 points) Find the minimal polynomial of 2 cos(27/9) over Q.
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