If |G| = pq, where p and q are primes that are not necessarily distinct,prove that |Z(G)| = 1 or pq.

Q: Let a, n E N. Assume that n > 2. Show that if Va is rational, then it is an integer.

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Q: Abstract Algebra. Answer in detail please.

A: To prove the statements about the given field extensions, (two quadratic and one 4th degree over Q)

Q: Let A = {n ∈ Z; n is odd} and B = {n ∈ Z; 3n 2 + 7 is even}. Prove that A = B.

A: We have to prove A is contain in B and vise versa

Q: O a) Prove by induction on nzi that 2*+s*.r. ... 3a-1)² = +n(6n*+3n -1). g'+ ... +(3n-1)? =…

A: To prove the given statements by using mathematical induction.

Q: For any n > 1, prove that the irreducible factorization over Z ofxn-1 + xn-2 + . . . +x +1 is π…

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Q: & Give an example of a polynomial (x) € Z[] and a prime p such that f(x) is reducible in Qx] but…

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Q: 3. x* + 1 is reducible over Z, for every prime p. a) True b) False

A: We have to find out whether x4+1 is reducible over Zp for every prime p or not a polynomial is said…

Q: suppose a prime p of the form 8n + 3 and that q = (p-1) / 2 is also prime. Prove that ordp (2) = 8n…

A: Consider the prime p is of the form 8n+3 and q=(p-1)/2 is also prime. Then, objective is to prove…

Q: Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if…

A: Given, r = a/b, where b = 2n5m for some n, m ≥ 0. We know, that any integer divided by 10N has a…

Q: d) If n iš a If t denotes the nth triangular number, prove that in terms of the binomial…

A: The binomial coefficient, nr is defined using the formula, nr=n!(n-r)! r! . The binomial expression,…

Q: What is the smallest positive integer k such that xk = e for all x inU(7 . 17)? Generalize to U(pq)…

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Q: Q2) If n is a prime number, then prove that Z/(n) is integral domain.

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Q: Show that a fraction r = a/ b in lowest terms has a finite decimalexpansion if and only ifb = 2n5m…

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Q: For m an n non-negative integers with 0 < m <n, give a combinatorial proof for the identity EOC) m…

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Q: Let p be a prime. Solve the equation x2 = 1 in Z/p (i.e. find all solutions, and prove that there…

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Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

Q: M + N = {m+n : m € M and n € N}. Prove that for all n € N, the number sup(M + N) − n is an upper…

A: Given:   If M and N are both bounded subsets of R. M+N :=m+n : m∈M  and  n∈N

Q: Let be a rational prime, with p = 1(mod 4). Prove that p is the product of two Gaussian primes that…

A: We will use the basic knowledge of number theory to answer this question properly and completely.

Q: (b) Prove that for any positive integer n, Ej. 2 = (n – 1)2"+1 + 2 (c) n(n² – 1) Prove that for any…

A: b) Let the statement be P(n): 1·21+2·22+3·23+⋯ +n·2n(, or,∑j=1nj·2j )=(n-1)2n+1+2. For n=1, P(1) :…

Q: For p an odd prime in Z , Prove: If p is not prime in Z[i] then -1 is a perfect square in Z/pZ

A: Given p is an odd prime in Z.Assume that (-1) is a perfect square in Z/pZ.Then, there exists an…

Q: For all n E N,n > 1, there exists a prime p such that p|n. ulgn hi

A: False

Q: 10. a.If f(t) = tm and g(t) = tn, where m and n are positive integers, show that…

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Q: Prove that for all integers m and n, if m is odd and the remainder is 2 when n is divided by 3, then…

A: Given, m is odd  When n is divided by 3, then the remainder is 2  We have to prove when  3m+2n  is…

Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p is not prime in Z[i]

A: Given that p is odd prime. This implies that p is not equal to 2.

Q: Let n E Z', where n > I. Show that if n is a product of distinct Mersenne primes, then o(n) = 2k for…

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Q: Q2: Let R a, b, c E Z}and let p: R → Z be defined such that = a. 4. Show that ker () is a prime…

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Q: ) Show that if p is a prime such that m +1< p < 2m + 1, then p divides (2m+1).

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Q: Let p and q be distinct primes with p<q and q# 1mod(p). Show that if G|= pq then G Z/pqZ.

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Q: For all n E N,n > 1, there exists a prime p such that p|n. صواب İhi O

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Q: Let p be prime. Prove that if H ≤ G, |G| = 2p and H is not normal, then |H| = 2.

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Q: Show that if n = p1p2⋯pk, where p1, p2,…, pk are distinct primes that satisfy pj − 1 | n − 1 for j =…

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Q: Let c be a number such that |c| 0. Then use the binomial theorem to show that 1 |c"| < 1+ nd 1 nd…

A: According to the given information, Let c be a number such that:

Q: Let m and n be positive integers and let k be the least common multipleof m and n. Show that mZ∩ nZ…

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Q: Let p be a prime. Show that xpn −x+1 ∈ Fp[x] is irreducible iff either (1) p = 2 and n =1,2 or (2) p…

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Q: there are no integers x and y such that x is a prime greater than 5 and x=6y+3

A: Given query is to check the statement whether it's possible or not.

Q: 4. Prove that f+ f+..+f = fnfn+1, where n is a positive integer and fn is the nth Fibonacci number.

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Q: Let n = pq where p and q are two distinct primes and consider Zn. Further, let [a] E Zn be an…

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Q: Take non-zero positive integers m and n. Show that Z/mZx Z/nZ≈ Z/mnZ if and only if m and n are…

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Q: Show that Zn has a nonzero nilpotent element if and only if n is divisibleby the square of some…

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Q: The Euler totient function is defined as ϕ (m) = |{k : 1 ≤ k ≤ m gcd(k, m) = 1}|,  or rather the…

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Q: Let m >1 be an integer, and let R = Z/mZ. Prove that R is nice if and only m is prime. Proof. ...

A: Given: m&gt;1 is an integer and R=Z/mZ. To show R is nice if and only m is prime.

Q: Suppose that j is prime and j2 | ab. Prove that if gcd(a, b) = 1, then j2 | a or p2 | b.

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Q: For every prime p, show that Х-1 — 1 %3 (х — 1)(х — 2) : .. [x — (р — 1)] in Zp[x].

A: Fermat's Little Theorem: Let p be a prime and let a∈N , 0&lt;a&lt;p. Let Ap={1a,2a,....(p-1)a} No…

Q: Let n = pq where p and q are two distinct primes and consider Zn. Further, let [a] E Zn be an…

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Q: Use mathematical induction to prove that for z>0 and any positive integer n, e21+z+ 2! n!

A: Recall that,    If f'(x)≤0 in [a,b] then f(b)≤f(a)  and   If f'(x)≥0 in [a,b] then  f(b)≥f(a)

Q: (c) Use Taylor's formula to find a positive integer n such that the error in the approximation…

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Q: Prove that if n = p1a1...prar, where the pi are prime, then σ(n)ϕ(n) = n2(1-p1-a,-1)...(1-pr-a,-1

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Q: For all n E N,n > 1. there exists a prime p such that p|n.

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Q: For every prime p, show that XP-1 – 1 = (x – 1)(x – 2) · · · [x – (p – 1)] in Z„[x].

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Q: Take non-zero positive integers m and n. Show that Z/mZx Z/nZZ/mnZ if and only if m and n are…

A: Given That:To take non-zero integers m and n .   ℤmℤ × ℤnℤ ≅ ℤmnℤ if m and n are relatively prime ,…