Let K = Q(ζ3) for a primitive cube root of unity ζ3 ∈ C Determine the splitting behaviour in K (remains prime, splits or ramifies) of the following primes: 2, 3, 5, 7, 11, 13.

Q: 1. Show that the square root of 3 is irrational as follows: (a) If 3p² = q² for p, q E Z, show that…

A: Let us prove the above result.

Q: (2) (b) (c) Use the Euclidean algorithm to determine the gcd of 85 Write down all the divisors of 85…

A: Let the given numbers are a = 85 and b = 135 (a)    From Euclidean algorithm.…

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Q: 3. Define primitive root and find all primitive root of following. (b) 19 (c) 15 25

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

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Q: Prove the following statements by contradiction. Theorem. (a) There do not exist integers x and y…

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Q: If p≥2 is a prime and a,b ∈Z(p) with a̸ = 0 ,we say that b/p a is the solution x∈Z(p) of the…

A: This is a question of Number Theory.

Q: a) If q(n) = byn* + Bk-1n*-1+...+ b1n+ bo is a polynomial in the variable ne N with be # 0, show…

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Q: (a) Show that the set Rof all multiples of 3 is a subring of 7. (b) Let k be a fixed integer. Show…

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Q: we wish to expand the bionomial expresjon (x-1) The bionomial theorem tells is that there will be 13…

A: Given, x-112  We have to find coefficient of x11 and x10  term.

Q: For the pair of integers 315 and 825: (a) Find the prime factorization of each number. (b) Find the…

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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: Find the smallest positive integer N≥ 2 that gives remainder 1 upon division by each of the integers…

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Q: ) Use the Chinese Remainder Theorem to find all solutions be- tween 0 and 1000 satisfying the…

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Q: 1) i) Prove that if a and b are integers, with b>0, then there exist unique integers q and r…

A: A question in elementary number theory. We shall solve this using basic facts of divisibility.

Q: 3. Prove that the cube roots of 1 are given by -1+i√√3 =3 2 || 2 -1-i√√3 w³ = 1. 2

A: DETAILED SOLUTION IS PROVIDED BELOWExplanation:

Q: For any integer k greater than or equal to 2, define R(k) as the ratio of k 's largest positive…

A: Let's break down the problem into two parts: (a) For every natural number 𝑥, there is a natural…

Q: Let p ∈ Z be a prime integer. (a) Show that if p > 3, then p is of the form 6k + 1 or 6k −1 for some…

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Q: Prove that every perfect square is either a multiple of 5 or of the form 5j - 1 or 5j + 1.

A: Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap −…