Let Z3[i] = {a + bi : a, b ∈ Z3, i2 = −1} where arithmetic is done mod 3. 1. Let α = 1 + i. Compute all the powers αk for k ∈ {1, 2, . . . , 8}. What can you say about Z3[i]∗?

Q: Let a, b be integers. What does it mean for a = b (mod 17)? Describe the elements of Z17. Compute…

A: a ≡ b (mod 17) means that a − b = 17k for some k ∈ ℤ. ∴  b − a = −17k = 17(−k); henceb ≡ a (mod…

Q: 8. Find the remainder of x2021 + x2020 + 1 modulo x2 – 1 in Q[x].

A: Remainder of the Polynomial

Q: 1. Show that k1601 = k (mod 17) for any k.

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Q: Show that x2 = Y mod p has

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Q: et x, y E Z. We have (x+y)³ = (x+y)² (x+y) = (x² +2xy+y?)(x+y) = (x³+3x?y+3xy² + y³) = x³ + y3 od 3.

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Q: For any a ∈ N, prove that a3 − a is divisible by 3. [Hint: Split into three cases: a = 3n, a = 3n +…

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Q: Let Z3[i] = {a + bi : a, b ∈ Z3, i2 = −1} where arithmetic is done mod 3. 1. What is |Z3[i]|? Give…

A: Given , Z3[i] = { a+bi : a , b ∈ Z3 , i2 = -1 } (.)  Zp[i] is a field if and only if p is a prime…

Q: 1. Prove (2p+1) = 2 (mod p) when p is any prime.

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Q: 4. Find the principal representative of each of the following: x³ -x + 1 mod x + 2 in Q[x] i. x3 ii.…

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Q: Prove that A=I-N+N²-N³ for A & N definec as follows: N=A-I (identity), where |1 0 0 an & I= 0 1 0. 0…

A: given matrix  A=1a12a1301a23001 and I=100010001 prove  that A-1=I-N+N2-N3 where N=A-I

Q: = a O a = a for all positive integers a. = a O b = b O a for all positive integers a and b. = If a >…

A: We apply the given formulas repeatedly.

Q: 61 Let A = Compute 612 - A and (612)A. 4 2

A: Step 1: Step 2: Step 3: Step 4:

Q: Show that the modulus of the quotient of tow complex conjugate numbers is 1, that is :1.

A: Use the following formulae, to prove the required result. The conjugate of the complex number a+ib…

Q: Find all 6 permutations of (12)(345).

A: Find all permutations with 12 in the first pair of bracket : (12)(345) (12)(354) (12)(435) (12)(453)…

Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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Q: Suppose n₁, n₂ are positive integers and gcd(01, n₂) = 1. Then for every integer x, if n₁x and n₂|x,…

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Q: 13. Find another representation 2, in which 3 a. ris negative and 0 < 0 < 2T; b. ris positive and…

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Q: (c) Suppose n EN is divisible by 5. What is (25n) (n) L?

A: Let, the prime factorization of n be as below.          n = p1a1 p2a2 . . . . . . pkak     for some…

Q: (a) Prove the following: i. 11 + (6 x 4") - 22n = 6 mod 7, 1 is divisible by 131, ii. 8260 iii.…

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Q: Prove that for any natural number n, 2(√√n +1 − 1) <1+ 1 √2 + <2√n.

A:

Q: (a) Find an r E Z that makes the following congruence true: 5x = 1 (mod 24). (b) Prove that there is…

A: Solution a: Given  5x≡1 mod 24 For x=1 5≡5 (mod 24) for x=2 10≡10 (mod 24) for x=3 15≡15 (mod 24)…

Q: 6. Let A = {nk Z, n = 6k+5} and B = {nn is odd}. Prove that AC B.

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Q: 2. List all elements of A = M₂ (Z₂). Give the units in A, as well as the zero divisors in A.

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Q: Let c, d e Z and m e N. For each statement decide whether it is necessarily true, or whether it can…

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Q: If a =b (mod m) then a" = b" (modm) Vn e Z*.

A: Given a≡bmod m. we have to show that an≡bnmod m. we wil prove this by result by using method of…

Q: Let E-f(): 05 05 n. 05 25 1.7√/25 rs 7). Compute 6.8365-1051090 11²³ dv

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Q: Let Z3[i] = {a + bi : a, b ∈ Z3, i2 = −1} where arithmetic is done mod 3. 1.  Is Z3[i] an integral…

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Q: Determine whether e(x)=15x+4(mod 26) and e(x)= 4x+5(mod 26) are invertibles or not. If yes, find the…

A: We are asked to determine if the given equation are invertible and if they are we have to find the…

Q: For each part, find the integer 0 ≤ r < |m| such that a ≡ r mod m: (a) m = 7, a = 5 (b) m = 5, a =…

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Q: Let

A: We construct multiplication table for Z3[i]