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 have the form b2 for any b in GF( p). Show that a is a nonsquare in GF(pn) if n is odd and that a is a square in GF( pn) if n is even

Q: Suppose that [E:Q] = 2. Show that there is an integer d such that E = Q(square root of d) where d is…

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Q: Show that if p is an odd prime, then every divisor of the Mersenne number 2p − 1 is of the form 2kp…

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Q: Let a,n∈N. Assume that n≥2. Show that if a^(1/n) is rational, then it is an integer.

A: The detailed solution is as follows below:

Q: Prove the square of any odd integer is expressible in the form 8n+1 with n ∈Z.

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Q: what are the prime ideals of Z2xZ2

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Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169.

A: Topic - principle of mathematical induction

Q: x4 >is ........in Q[x] Not prime ideal ONone of the choices O Prime not maximal

A: Answer is maximal ideal , i.e , last option , and proof is given below.

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

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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: Suppose that p is a prime and p ≠ 2. Let a be a nonsquare inGF(p)—that is, a does not have the form…

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

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Q: Use a direct proof technique to prove the following theorems: For all integers z and y, r + y-3 is…

A: Solution is in step 2

Q: Prove that if p is a prime, then the square root of p is irrational.

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Q: b) Suppose that A is countably infinite and that there is a one-to-one corresp between A and B.…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Prove that if n is odd, then 2 is not a square in GF(5n).

A: Given n is odd number We have to show that 2 is not a square in GF5n. We know that the field GF5 is…

Q: Given the polynomials P1 10101 and P2= 11000. Show the result of multiplying P1 P2 in GF(25) using…

A: Given the polynomial P1=10101 and P2=11000. Show the result of multiplying P1⊗P2 in GF25 using the…

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: ivided by 6.

A: Given, m is odd. So, m = 2k+1 for some integer k  &, n leaves remainder 2 when divided by 3  So,…

Q: Prove that 67n+ 17 cannot be a perfect square for any three consecutive integers n.

A: We will use quadratic residue and proof by contradiction method.

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: Let m be a positive integer and suppose that the only prime divisors of m are 2, 3, 5 and 11. Show…

A: Given: Let m be a positive integer and suppose that the only prime divisors of m are 2, 3, 5 and 11.…

Q: Prove that if n is odd, then 2 is not a square in GF(5n) In other words, prove that there is no…

A: PROOF :

Q: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p

Q: Prove that the statement is true for every positive integer n.3 is a factor of n3 + 2n

A: Consider the given statement, "3 is a factor of  n3+2n" for all positive integer n Suppose Pn…

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: S Prove by mathematical induction that x 1 is a factor of x" 1 for all integers n > 1.

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Q: Prove that if n is odd, then 3 is not a square in GF(5n).  In other words, prove that there is no…

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Q: . Use direct proof to prove that, if x is a positive odd integer, then z – 1 is wholly dividable у…

A: x^2-1 is divisible by 4.

Q: Prove that n² +n is divisible by 2, where n E N.

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Q: Use proof by cases to prove that for any integer n, n² + 3n + 7 is even.

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Q: Use a direct proof technique to prove the following theorems: For all integers z and y, r² + y - 3…

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Q: Prove that 3 divides n3+2n whenever n is a positive integer

A: We use mathematical induction. Let P(n) be "3 divides n3+2n" For n=1 n3+2n=(1)3+2(1)=3 We then note…

Q: Show that if p is an odd prime, then there are exactly (p − 1)∕2 quadratic residues of p among the…

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Q: Prove that n2+n  is  divisible by 2, where nN.

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Q: Suppose a is a positive integer and p is a prime. Prove that pla if and only if the prime…

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Q: Prove that n2 + 4 is not divisible by 3 for all integers n.

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Q: If |G = 30 and |Z(G)| that |G| 5, what is the structure of G/Z(G)? What is the structure of G/Z(G)…

A: Solution:-

Q: Suppose that [E:Q] = 2. Show that there is an integer d such thatE = Q(√d) where d is not divisible…

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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: Show that any integer of the form 6k + 5 is also of the form 3j + 2, but not conversely.

A: 2. We can write 6k+5 = 6k+3+2=3(2k+1)+2 lets write 2k+1=j hence 6k+5=3j+2

Q: Suppose p and q are distinct primes such that ordp a = h and ordq a = k Prove that ordpq a= lcm(h,…

A: Here we use basic group theory.

Q: Determine whether each of the following is true or false. Prove each true statement, and give a…

A: Read the statement very carefully. Now use the properties and definition to check whether the…

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

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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: 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: For any integer n2 1. prove that: (a) 1+2+3+..+ (n – 1) + n + (n – 1) + . .. +3+2+ 1 = n². (b) +...+…

A: Note: According to Bartleby guidelines; for more than three questions asked, only the first three…

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

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.