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

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)

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Q: () Prove by induction that for any positive integer n, n(n +1)(2n + 1) j=1

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Q: 6) The Ying of integers(2) is ideal of veal number (R) an DTrue D Fulse

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Q: For integer n>1, nZ={nx:x in z}, nZ is set of all integer multiples of n. Prove that integer p is…

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Q: Let p be a prime. Show that there is exactly two elements a belongs to Zp s.t a^2 = 1.

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Q: Prove by induction that for positive integers n: E (312 + 2i + 5) = }(2n³ + 5n² + 13n)

A: To prove the given statement by mathematical induction.

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…

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Q: Let be a prime such that p = 1 (mod 4). Prove that –1F, is a square in Fp. -

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Q: Prove, by induction, that, for n2 5, the nth Fibonacci number satisfies Fn s 2".

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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: Q2: Prove by Induction that Ŝ 2i+1 = h z0 ( The som of the fist had positive odd integers fum is…

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Q: For p an odd prime in Z , Prove: If -1 is a perfect square in Z/pZ then p= 1(mod 4)

A: This can be done as follows:

Q: Let a, b € Z with a | b. Prove that an | bn for every positive integer n.

A: Given that           a, b∈ℤ with a|b. To prove an|bn  , we shall use mathematical induction.

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: (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: (b) Prove that for any positive integer n, j. 2³ = (n – 1)2"+1 +2 j=1 (c) n(n? – 1) - Prove that for…

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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: Let p be a prime divisor of the Fermat number Fn 22" + 1. : 2n+1 (a) Show that ord, 2 (b) Use…

A: As per the question we have the Fermat number Fn = 22ⁿ+1 and its prime divisor p. Here we will use…

Q: 13. Prove by Math Induction e* > 1 + x for all positive integers x > 1

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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: (e) Prove that for any non-negative integer n, i 3 n - 3"+1 – (n+ 1)3" + 1] 4 j=0 (f) Prove that for…

A: (e)…

Q: 7. Prove by induction. For any n2 0, E F = F,Fn+1• k=0

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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: Prove that I +:+ + for every positive integer n.

A: In this question, principle of mathematical induction is used. Principle of Mathematical Induction…

Q: Q1. Let x and y be two integers. Prove that for all integers m 2 1, x™ – y™ is a multiple of x-y…

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Q: Let p e Z be prime. Prove that there do not exist nonzero a, b e Z+ such that a2 = pb2

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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: In the domain of Gaussian integers Z[i], the element 15 is 青 O Reducible O Unit O Irreducible O None…

A: Since 3∈ Z[i] and 5∈ Z[i] And 15 = 3 × 5 Hence 15 is reducible element in Z[i]

Q: Prove or disprove: there exists a E N such that for all n E N, an + 1 is prime.

A: According to the given information, it is required to prove or disprove:

Q: Find (with proof) all positive primes p such that 217p is a Carmichael number.

A: Carmichael number - A number n is a Carmichael number if and only if n=p1p2…pj , a product of at…

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 p e Z be prime. Prove that there do not exist nonzero a, b e Z+ such that a2 = pb2.

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Q: Find the integral basis and discriminant of the number field K = Q(/=p), where p is a prime such…

A: It is given that

Q: Let a and b be positive integers and let d = gcd(a, b) and m =lcm(a, b). If t divides both a and b,…

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Q: Given positive integers k and m, there exists an integer N ≥ m such that

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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: 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: If m, n are relatively prime, then o(mn) = o(m)o(n)

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Q: 20. If p(x) and g(x) are nonassociate irreducibles in Flx), prove that p(x) and g(x are relatively…

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Q: Let n be a positive integer. Prove that for all x € Z, if x" is odd, then x is odd.

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

A: Given: G=p.q where p & q are primes To prove: ZG=1 or pq

Q: 2 Lk=0 4. Prove that for all positive integers n, п

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Q: Prove that for any positive integer n, Vn is either an integer or irrational.

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Q: -- a) such that ab =1. Let p>1 be a prime number. Show that for each a + 0 in Zn there is b in Zp…

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Q: 1. Find the least no EN such that 10k + 3 no, it has the stated property. and prove that

A: Given: 10k+3<2k, ∀k≥n0, where n0∈ℕ is a least one. To find: Value of n0. To prove: 10k+3<2k,…

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 ,…