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

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: Find the principal ideal (z) of Z such that each of the following sums as is equal to (z).   (4) +…

A: Known fact: m+n=d    where d=gcdm, n

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: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p= 1 (mod 4)

A: Compute general prime that satisfies given condition as follows.

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: Let p be a prime divisor of the Fermat number Fn= 22^n+ 1 i. Show that ordp2 = 2n+ 1 ii. Use…

A: Given p be a prime divisor of the number Fn=22n+1. (i) Since p divides Fn=22n+1, therefore: Fn ≅0…

Q: 7n+1 20n+3 is irreducible for every natural number n. Prove that the fraction

A: Solution is given below:

Q: |State whether the following statements are true or false. Show working. a. For each odd prime p, a…

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Q: Exercise 3: Suppose V2 E Q (i.e V2 is a rational number). p p Therefore, there exist p, q E Z, q +…

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Q: If p > 2 is a prime of the form 2m +1, prove that m is a power of 2.

A: Let p be a prime and p > 2. And, p = 2m +1 Suppose 't' is an odd prime factor of m. That is, m =…

Q: (b) Prove that -1+²+...+x+ 1 is irreducible over Z if and only if n is a prime

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Q: (7) In the ring (Z, +,.), we get n {P: P non trival prime ideal in Z} %3D ....... (a) o (b) (Z, +,.)…

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Q: Find all the positive prime p such that 9p+1 is a perfect cube. Namely, such that there exists an…

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Q: Find a positive integer m > 3 such that there are at least 3 integers n in Zm such that n2 = 1.

A: We have to find a positive integer m≥3 such that there are at least 3 integer n in Zm such that n2=1…

Q: |State whether the following statements are true or false. Show working. a. For each odd prime p, a…

A: We have to state whether the statements are true or false

Q: Find all the positive primes p such that 9p+ 1 is a perfect cube. Namely, such that there exists an…

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Q: The ideal generated by (1,0) is prime in Z×Z. Select one: O True O False

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

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Q: 1. Let p be a prime. Let E:= {a € Z, : aP - a = 0}. Show that E is a splitting AND separable…

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Q: Find the supremum and infimum of the set A := ∩ ∞ n=1 (1 −1/n ,1 +1/n ). And prove it

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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: 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: (ii) Prove, by induction, that F, × F, × F, × ...× F-1 = F, – 2. %3D (iii) Use part (ii) to prove…

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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: (a) Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i). (b) Show…

A: Consider the given information: To show that 2 is equal to the product of a unit and the square of a…

Q: Find all the prime ideals and all maximal ideals of 7e:

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Q: Prove that for all primes p2 5, p + 1 cannot be a perfect square. Recall that is a perfect square if…

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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: Factor 19 into irreducible elements in the Gaussian integers, Z[i].

A: We have to factor 19 into irreducible elements in the Gaussian integers, ℤ[i]. Irreducible element :…

Q: a. Suppose p is an odd prime, and let A := a, b € ? (a) Suppose there are ao, bo € Z such that p =…

A: Given p is an odd prime and  A=ab-ba:a,b∈ℤp Suppose, there are a0,b0∈ℤ such that p=a02+b02. Then, by…

Q: Use Fermat's theorem to show that for any two distinct odd primes P.9, and any integer C: CM-c-c +c…

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Q: ) Prove that for all positive integers k, 22k – 1 is a multiple of 3.

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Q: Let n = pq be the product of two odd primes. Recall that there are no primitive roots modulo n. Show…

A: We use the idea that if primitive root exists then x^2 = 1 mod (m) has only 2 solutions. The…

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: (2) Let p be an odd prime and suppose that m e Z is not divisible by p. (1) Show that E() \p- = 0.…

A: (1) Given that p is an odd prime. The aim is to show that ∑a=1p-1ap=0. Now take, ∑a=1p-1ap,…

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

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Q: 2. (a) Define the Euler phi-function and use its properties from the lectures to prove that for all…

A: Euler's phi function is one of the most important number theoretic functions. For any positive…

Q: Let p be a prime, a ∈ Z. Prove that ap − a is divisible by p.

A: Fermat's Theorem : If p is any prime and a is any integer then p divides (ap-1-1).

Q: 10 a. Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i). b.…

A: We will give the result in next step. Note that the ring given is called ring of Gaussian integers.

Q: Let be an odd prime. Prove that for all i with 1 sisp-2 one has p| 1° + 2' + ... + (p – 1)'.

A: Given : p is an odd prime To Prove :          p| 1i+2i+. . .+p-1i for all i with 1≤i≤p-2

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: 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: For integers m, n, p, q E Z, n, q0, use this and the field axioms (a) Prove m + P 9 mq+np and m. 2…

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Q: 1) Let p be a prime number. Show that Z) := {E Q| p does not divide b is a subring of Q with only…

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Q: Let x, y, A, µ E Z. Prove that if 1 = Ax + µy then x, y are relatively prime. a (a) Find a prime…

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Q: Suppose that p is a prime and p does not equal 2. Let a be a nonsquare in GF( p)—that is, a does not…

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Q: Determine the type (Inert, Ramified or Spilt) of the following G/Eprimes by plotting all the…

A: Given

Q: Prove that if p is a prime in Z that can be written in the form a2 + b2,then a + bi is irreducible…

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Q: Prove that if D is a PID and p ∈ D is irreducible, then p is a prime of D

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Q: (7) In the ring (Z,+,.), we get n{P:P non trival prime ideal in Z} = ...... (a) o (b) (Z, +, .) (c)…

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