Question

In: Advanced Math

11.4 Let p be a prime. Let S = ℤ/p - {0} = {[1]p, [2]p, ....

11.4 Let p be a prime. Let S = ℤ/p - {0} = {[1]p, [2]p, . . . , [p-1]p}. Prove that for y ≠ 0, Ly restricts to a bijective map Ly|s : S → S.

11.5 Prove Fermat's Little Theorem

Solutions

Expert Solution


Related Solutions

Let p be an integer other than 0, ±1. (a) Prove that p is prime if...
Let p be an integer other than 0, ±1. (a) Prove that p is prime if and only if it has the property that whenever r and s are integers such that p = rs, then either r = ±1 or s = ±1. (b) Prove that p is prime if and only if it has the property that whenever b and c are integers such that p | bc, then either p | b or p | c.
let R = Z x Z. P be the prime ideal {0} x Z and S...
let R = Z x Z. P be the prime ideal {0} x Z and S = R - P. Prove that S^-1R is isomorphic to Q.
2. (a) Let p be a prime. Determine the number of elements of order p in...
2. (a) Let p be a prime. Determine the number of elements of order p in Zp^2 ⊕ Zp^2 . (b) Determine the number of subgroups of of Zp^2 ⊕ Zp^2 which are isomorphic to Zp^2 .
Let p be a prime and d a divisor of p-1. show that the d th...
Let p be a prime and d a divisor of p-1. show that the d th powers form a subgroup of U(Z/pZ) of order (p-1)/d. Calculate this subgroup for p=11, d=5; p=17,d=4 ;p=19,d=6
2. Let X ~ Geometric (p) where 0 < p <1 a. Show explicitly that this...
2. Let X ~ Geometric (p) where 0 < p <1 a. Show explicitly that this family is “very regular,” that is, that R0,R1,R2,R3,R4 hold. R 0 - different parameter values have different functions. R 1 - parameter space does not contain its own endpoints. R 2. - the set of points x where f (x, p) is not zero and should not depend on p. R 3. One derivative can be found with respect to p. R 4. Two...
Let Xn is a simple random walk (p = 1/2) on {0, 1, · · ·...
Let Xn is a simple random walk (p = 1/2) on {0, 1, · · · , 100} with absorbing boundaries. Suppose X0 = 50. Let T = min{j : Xj = 0 or N}. Let Fn denote the information contained in X1, · · · , Xn. (1) Verify that Xn is a martingale. (2) Find P(XT = 100). (3) Let Mn = X2 n − n. Verify that Mn is also a martingale. (4) It is known that...
Let A =   [  0 2 0 1 0 2 0 1 0 ]  . (a)...
Let A =   [  0 2 0 1 0 2 0 1 0 ]  . (a) Find the eigenvalues of A and bases of the corresponding eigenspaces. (b) Which of the eigenspaces is a line through the origin? Write down two vectors parallel to this line. (c) Find a plane W ⊂ R 3 such that for any w ∈ W one has Aw ∈ W , or explain why such a plain does not exist. (d) Write down explicitly...
Let V = {P(x) ∈ P10(R) : P'(−4) = 0 and P''(2) = 0}. If V=...
Let V = {P(x) ∈ P10(R) : P'(−4) = 0 and P''(2) = 0}. If V= M3×n(R), find n.
Let p be an odd prime (i.e., any prime other than 2). Form two vector spaces...
Let p be an odd prime (i.e., any prime other than 2). Form two vector spaces V1, V2 over Fp (prime field of order p) with bases corresponding to the edges and faces of an icosahedron (so that V1 has dimension 30 and V2 has dimension 20). Let T : V1 → V2 be the linear transformation defined as follows: given a vector v ∈ V1, T(v) is the vector in V2 whose component corresponding to a given face is...
Let p be the prime number (2^20)*(3^7)5 + 1 = 11466178561. Solve for x such that...
Let p be the prime number (2^20)*(3^7)5 + 1 = 11466178561. Solve for x such that 2^x ≡ 2376886429 (mod p) Explain your method carefully.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT