Let p be a prime >0. Prove that 12,23 (21) gives a set of different remainders modulus p. Also prove that for every number a with pla, a is congruent to one and only one of the element in the...
please solve this. (number theory) Suppose that p is a prime. Prove that pla if and only if pla?.
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...
Please prove the 3 theorems, thank you! 7.6 Theorem. Let p be a prime. Then half the numbers not congruent to 0 modulo p in any complete nesidue system modulo p are quadratic residuess modulo p and half are quadratic non-residues modulo p. From clementary school days, we have known that the product of a pos- itive number and a positive number is positive, a positive times a negative is negative, and the product of two negative numbers is positive....
A 13. Let X be a p-element set and let Y be a k-element set. Prove that the number of functions f :X >Y which map X onto Y equals k!S(p, k) S#(p, k) : A 13. Let X be a p-element set and let Y be a k-element set. Prove that the number of functions f :X >Y which map X onto Y equals k!S(p, k) S#(p, k) :
please prove proofs and do 7.4 7.2 Theorem. Let p be a prime, and let b and e be integers. Then there exists a linear change of variahle, yx+ with a an integer truns- farming the congruence xbx e0 (mod p) into a congruence of the farm y (mod p) for some integer 8 Our goal is to understand which integers are perfect squares of other inte- gers modulo a prime p. The first theorem below tells us that half...
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the set S defined as follows: Va,bE S, arb if and only if every prime number that divides a is a factor of b and a S b. The relation T is a partial order relation (you do not need to prove this). Draw the Hasse diagram for T 1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the...
It is important.I am waiting your help. 11. a) Prove that every field is a principal ideal domain. b) Show that the ring R nontrivial ideal of R. fa +bf2a, b e Z) is not a field by exhibiting a 12. Let fbe a homomorphism from the ring R into the ring R' and suppose that R ker for else R' contains has a subring F which is a field. Establish that either F a subring isomorphic to F 13....
Please prove the theorems, thank you 6.1 Theorem. Let anx+an-1- +ag he a polynomial of degree n0 with integer coefficients and assume an0. Then an integer r is a Poot of (x) if and only if there exists a polynomlal g(x) of degree n - with integer coeficients such that f(x) (x)g(x). This next theorem is very similar to the one above, but in this case (xr)g(x) is not quite equal to f(x), but is the same except for the...
8. Let p be an odd prime. In this exercise, we prove a famous result that characterizes precisely when -1 has a sqare root 1 mod 4. (You will need Wilson's Theorem for one (mod p). Prove: a 2--1 mod p has a solution if and only if p dircction of the proof.) 8. Let p be an odd prime. In this exercise, we prove a famous result that characterizes precisely when -1 has a sqare root 1 mod 4....