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, L_y restricts to a bijective map L_y|_s : S → S.

11.5 Prove Fermat's Little Theorem

Expert Solution

Colby Messinger answered 2 years ago

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 gcd(a, p) = 1 with p a prime. Show that if a has at least...

Let gcd(a, p) = 1 with p a prime. Show that if a has at least one square root, then a has exactly 2 roots. [hint: look at generators or use x^2 = y^2 (mod p) and use the fact that ab = 0 (mod p) the one of a or b must be 0(why?) ]

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 S = {-3, -2, -1, 0, 1, 2, 3}. Define a relation R on S...

Let S = {-3, -2, -1, 0, 1, 2, 3}. Define a relation R on S by: xRy if and only if x = y + 4n for some integer n. a) Prove that R is an equivalence relation. b) Find all the distinct equivalence classes of R.

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.

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