10. Let [n] be an element in Zp, p prime. We say [n] is perfect provided [o (n)] [2n]. Show that ...
Problem 4. Let p be an odd prime, and let Tp C Zp denote the set of elements of Zp which are perfect cubes: Tp-(a: a E z;} (1) Show that if p1 (mod 3) then Tp (p 1)/3. Problem 4. Let p be an odd prime, and let Tp C Zp denote the set of elements of Zp which are perfect cubes: Tp-(a: a E z;} (1) Show that if p1 (mod 3) then Tp (p 1)/3.
(1) Let p be a prime number. The following polynomials are considered as elements in Zp[ (a) Show that zP-1-(z -1)( 2) ( (p 1)) (b) Let φο : Zp[2] Zp be the evaluation homomorphism at 0. Compute φο(zp-1-1) and φο((1-1)(1-2) . . . (z-(p-1))) (c) Use parts (a) and (b) to conclude that (-1)--1. (1) Let p be a prime number. The following polynomials are considered as elements in Zp[ (a) Show that zP-1-(z -1)( 2) ( (p 1))...
Let pEN be a prime and let V be a vector space over Zp with dimension n. Show that |v| = pn. Let pEN be a prime and let V be a vector space over Zp with dimension n. Show that |v| = pn.
7. EXTRA CREDIT [5 points] For Zp, where p is a prime, Fermat's theorem gives us an alternative way to compute the multiplicative inverse of any given nonzero [x]p: raise it to the power of p-1. Show that Fermat's theorem is a corollary (a special case) of Euler's theorem, i.e., show how one can derive the former from the latter.
12 ) - 2. Let p(n) denote the number ofdstinct prime divisors ofn. For example, p( p(24)-2 and p(60) 3. Let q(n)an, where a is fixed and show that qn) is multiplicative, but not completely multiplicative. 12 ) - 2. Let p(n) denote the number ofdstinct prime divisors ofn. For example, p( p(24)-2 and p(60) 3. Let q(n)an, where a is fixed and show that qn) is multiplicative, but not completely multiplicative.
8. Let p be a prime number. Define -c0t}cQ ZAp) Prove that Zp) is a subring of Q Prove that Z is a subring of Z Show that the field of fractions of Zp) is isomorphic to Q
Let p be a prime. Show that Zp(X)/(X2+1) is a field iff the equation x2=-1 has no solution (mod p).
10. Let p be a prime number. We know that p divides (p- 1)!+1. Show that if p> 5 then (p- 1)!+1 is never of the form pë where e e Z0 10. Let p be a prime number. We know that p divides (p- 1)!+1. Show that if p> 5 then (p- 1)!+1 is never of the form pë where e e Z0
Thee part question. Please answer all parts! Let E be a field of characteristic p > 0 (we proved p must always be prime). Verify that the ring homomorphism X : Z → E determined by sending χ : 1-1 E (the unity in E) ( so x(n)-n 1E wheren1E 1E 1E (n-times), x(-n)- nle for any n 1,2,3,... and X(0) 0E by definition of χ) is in fact a ring homomorphism with ker(X) = pZ. Úse the fundamental homomorphism...
(4) (10 points) Show that 3 is a prime element in Zg]. Find the irreducible Z8]. Specify the irreducible factors that appear in the factorization of 9t. ation of 9i in let Prime P-(3) Thus 3 divide s Nea) ar 3 divides NC) ud we can sahat divides N(O)x possibility t, Hat bof ,, hak rosidue omod 3 Henle 9ie3 and p.3 (s prime. 3, us Sethat we are left wl oor 1 , So the only if you work...