Let k > 3. Show that (1) 3 has order 2^(k-2) modulo 2^k . (2) {3, -1} is a generating set for 2^k


Let k > 3. Show that (1) 3 has order 2^(k-2) modulo 2^k . (2) {3,...
Let k > 3. Show that
(1) 3 has order 2^(k-2) modulo 2k .
(2) {3, -1} is a generating set for 2k .
Let k 3. Show that (1) 3 has order 2-2 modulo 2* (2) {3,- is a generating set for 2
Let k 3. Show that (1) 3 has order 2-2 modulo 2* (2) {3,- is a generating set for 2
(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.
3. Let p>3 be an odd prime and let {ri,r2, .r} be the set of incongruent primitive roots modulo p. Compute the product rir .r modulo p. Recall the proof of Wil- son's Theorem for inspiration
3. Let p>3 be an odd prime and let {ri,r2, .r} be the set of incongruent primitive roots modulo p. Compute the product rir .r modulo p. Recall the proof of Wil- son's Theorem for inspiration
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...
9. In Z/31Z, using Proposition 3, find a primitive root modulo 31. Proposition 3. Let a,b be elements of a finite abelian group. If a has order r, and b has order s, and (r, s) = 1, then ab has order rs.
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...
Let p be an odd prime and a an integer with p not dividing a. Show that a(p-1)/2 is congruent to 1 mod p if and only if a is a square modulo p and -1 otherwise. (hint: think generators)
Let KQi, 2 (a) Show that K is a splitting field of X4- 2 over Q. (b) Find a Q-basis of K c) Find an automorphism of order four of K over i (d) Determine all the automorphisms of K over Q (e) The zeros of X4-2 form -(±Vitiy2). Describe the action of the set S Aut(K) on S (f) Find all subgroups of Aut (KQ). (g) Find all intermediate field extensions of C K.
Let KQi, 2 (a) Show...
Let a and b be elements of a group G such that b has order 2 and
ab=ba^-1
12. Let a and b be elements of a group G such that b has order 2 and ab = ba-1. (a) Show that a” b = ba-n for all integers n. Hint: Evaluate the product (bab)(bab) in two different ways to show that ba+b = a-2, and then extend this method. (b) Show that the set S = {a”, ba" |...