1.32. Recall that g is called a primitive root modulo p if the powers of g...
1.32. Recall that g is called a primitive root modulo p if the powers of g give all nonzero elements of F p .
(b) For which of the following primes is 3 a primitive root modulo p ?
(i) p = 5 (ii) p = 7 (iii) p = 11 (iv) p = 17
Homework Answers
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1.32. Recall that g is called a primitive root modulo p if the
powers of g...
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...
3. The number 2 is a primitive root modulo 19; the powers of 2 modulo 19...
3. The number 2 is a primitive root modulo 19; the powers of 2 modulo 19 are listed below 21 22232425 26 228221212213214215216217 21 2 48 16 13 7 14918 175 36 125 101 Use this table to solve r 7 mod 19.
Let g be a primitive root modulo to the odd prime p. Prove that:
Let g be a primitive root modulo to the odd prime p. Prove that: 2)=-1 2)=-1
2.5. Let p be an odd prime and let g be a primitive root modulo has...
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 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...
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...
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...
7.23 Theorem. Let p be a prime congruent to 3 modulo 4. Let a be a natural number with 1 a< p-1. Then a is a quadrut...
7.23 Theorem. Let p be a prime congruent to 3 modulo 4. Let a be a natural number with 1 a< p-1. Then a is a quadrutic residue modulo pif and only ifp-a is a quadratic non-residue modulo p. 7.24 Theorem. Let p be a prime of the form p odd prime. Then p 3 (mod 4). 241 where q is an The next theorem describes the symmetry between primitive roots and quadratic residues for primes arising from odd Sophie...
(1) The Legendre symbol and Euler's criterion. (1 pt each) Let p be an odd prime and a Z an integ...
(1) The Legendre symbol and Euler's criterion. (1 pt each) Let p be an odd prime and a Z an integer which is not divisible by p. The integer a is called a quadratic residue modulo p if there is b E Z such that a b2 (p), i.e., if a has a square root modulo p. Otherwise a is called a quadratic non-residue. One defines the Legendr symbol as follows: 1 p)=T-i if a is a quadratic residue modulo...
Homework 19. Due April 5. Consider the polynomial p(z) = r3 + 21+1. Let F denote the field Q modu...
Homework 19. Due April 5. Consider the polynomial p(z) = r3 + 21+1. Let F denote the field Q modulo p(x) and Fs denote the field Zs[r] modulo p(x). (i) Prove that p(x) is irreducible over Q and also irreducible over Zs, so that in fact, F and Fs are fields (ii) Calculate 1+2r2-2r + in HF. (iii) Find the multiplicative inverse of 1 +2r2 in F. (iv) Repeat (ii) and (iii) for Fs. (v) How many elements are in...
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...
3. The number 2 is a primitive root modulo 19; the powers of 2 modulo 19 are listed below 21 22232425 26 228221212213214215216217 21 2 48 16 13 7 14918 175 36 125 101 Use this table to solve r 7 mod 19.
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 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...
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...
7.23 Theorem. Let p be a prime congruent to 3 modulo 4. Let a be a natural number with 1 a< p-1. Then a is a quadrutic residue modulo pif and only ifp-a is a quadratic non-residue modulo p. 7.24 Theorem. Let p be a prime of the form p odd prime. Then p 3 (mod 4). 241 where q is an The next theorem describes the symmetry between primitive roots and quadratic residues for primes arising from odd Sophie...
(1) The Legendre symbol and Euler's criterion. (1 pt each) Let p be an odd prime and a Z an integer which is not divisible by p. The integer a is called a quadratic residue modulo p if there is b E Z such that a b2 (p), i.e., if a has a square root modulo p. Otherwise a is called a quadratic non-residue. One defines the Legendr symbol as follows: 1 p)=T-i if a is a quadratic residue modulo...
Homework 19. Due April 5. Consider the polynomial p(z) = r3 + 21+1. Let F denote the field Q modulo p(x) and Fs denote the field Zs[r] modulo p(x). (i) Prove that p(x) is irreducible over Q and also irreducible over Zs, so that in fact, F and Fs are fields (ii) Calculate 1+2r2-2r + in HF. (iii) Find the multiplicative inverse of 1 +2r2 in F. (iv) Repeat (ii) and (iii) for Fs. (v) How many elements are in...
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