Problem 2. Let me N so that Zm has a primitive root. Show that it has...
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...
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...
fekri/n k0,1,...,n-1}, called the nth roots of unity. A primitive root of unity is = eri/n for which 2. The roots off(x) = x"-1 are the n complex numbers Cn and are ged(n, k) 1. It is easy to see that Q(C) is the splitting field of zn - 1. (a) For each n 3,... ,8, sketch the nth roots of unity in the complex plane. Use a different set of axes for each n. Next to each root, write...
8. Let g be a primitive root of an odd prime p, and suppose that p3 (mod 4). Show that -g is not a primitive root of p. 8. Let g be a primitive root of an odd prime p, and suppose that p3 (mod 4). Show that -g is not a primitive root of p.
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.
2.5. Let p be an odd prime and let g be a primitive root modulo has a square root modulo p if and only if its discrete logarithm log,(a) mod p. Prove t that is even.
Let p 37, and let g-5, which is a primitive root. Let h 31. We will use the baby-step giant step method with N 6 (which satisfies N p-1). (a) Compute g,g,g.g( p) (b) Compute hgo, hg , hg 1,hg18, hg24, hg30 (mod p). (c) Determine Lg(h) by spotting the match between these two lists.
Need help!! Please help — crypto math 1. Determine L13(18) for p 19. 2. Let p be prime, and α a primitive root mod p. Prove that α(p-1)/2-_1 (mod p). 3. It can be shown that 5 is a primitive root for the prime 1223. You want to solve the discrete logarithm problem 53 (mod 1223). You know 3611 Prove it. 1 (mod 1223). Is x even or odd? 1. Determine L13(18) for p 19. 2. Let p be prime,...
Problem 6. The number 65537 is prime, and 3 is a primitive root. Show what computation would prove 65537 is prime. You do NOT actually have to evaluate any multiplications or finding remainders, just show what the final result of the computation will be, and how many multiplications / divisions it would take.
no coding solve it by hand (2) Homogeneous Linear Recurrences where p(A) has repeated roots (a) Let Let f(n) = an. Show f(n) = d,2n +d2n2" satisfies a(ai)iez be a sequence of real numbers p(A)f(n) (A 2)2(f(n)) 0 for every di, d2 0. . (2) Homogeneous Linear Recurrences where p(A) has repeated roots (a) Let Let f(n) = an. Show f(n) = d,2n +d2n2" satisfies a(ai)iez be a sequence of real numbers p(A)f(n) (A 2)2(f(n)) 0 for every di, d2...