(a) Let n be odd and ω a primitive nth root of 1 (means that its...

(a) Let n be odd and ω a primitive nth root of 1 (means that its order is n). Show this implies that −ω is a primitive 2nth root of 1. Prove the converse: Let n be odd and ω a primitive 2nth root of 1. Show −ω is a primitive nth root of 1. (b) Recall that the nth cyclotomic polynomial is deﬁned as Φn(x) = Y gcd(k,n)=1 (x−ωk) where k ranges over 1,...,n−1 and ωk = e2πik/n is a primitive nth root of 1. Compute Φ8(x) and Φ9(x), writing them out with Z coeﬃcients. Show your steps.

Expert Solution

Colby Messinger answered 2 years ago

Let p be an odd prime. (a) (*) Prove that there is a primitive root modulo...

Let p be an odd prime. (a) (*) Prove that there is a primitive root modulo p2 . (Hint: Use that if a, b have orders n, m, with gcd(n, m) = 1, then ab has order nm.) (b) Prove that for any n, there is a primitive root modulo pn. (c) Explicitly find a primitive root modulo 125. Please do all parts. Thank you in advance

Number Theory: Let p be an odd number. Recall that a primitive root, mod p, is...

Number Theory: Let p be an odd number. Recall that a primitive root, mod p, is an integer g such that gp-1 = 1 mod p, and no smaller power of g is congruent to 1 mod p. Some results in this chapter can be proved via the existence of a primitive root(Theorem 6.26) (c) Given a primitive root g, and an integer a such that a is not congruent to 0 mod p, prove that a is a square...

An element a in a field F is called a primitive nth root of unity if...

An element a in a field F is called a primitive nth root of unity if n is the smallest positive integer such that an=1. For example, i is a primitive 4th root of unity in C, whereas -1 is not a primitive 4th root of unity (even though (-1)4=1). (a) Find all primitive 4th roots of unity in F5 (b) Find all primitive 3rd roots of unity in F7 (c) Find all primitive 6th roots of unity in F7...

Find a primitive root for: (a) n = 18, (b) n = 50 (c) n =...

Find a primitive root for: (a) n = 18, (b) n = 50 (c) n = 27, (d) n = 625.

Let P(n) := ” If n^3 is odd then n is also odd.” I.e., if ∃k...

Let P(n) := ” If n^3 is odd then n is also odd.” I.e., if ∃k ∈ Z, n3 = 2k + 1, ∃b ∈ Z, n = 2b + 1 a) Prove P(n) by contraposition b) Prove P(n) contradiction c) Prove P(n) using induction

x(n)=(1/4)|n|, find x(ω)

3. To begin a proof by contradiction for “If n is even then n+1 is odd,”...

3. To begin a proof by contradiction for “If n is even then n+1 is odd,” what would you “assume true? 4. Prove that the following is not true by finding a counterexample. “The sum of any 3 consecutive integers is even" 5. Show a Proof by exhaustion for the following: For n = 2, 4, 6, n²-1 is odd 6. Show an informal Direct Proof for “The sum of 2 even integers is even.” Recursive Definitions 7. The Fibonacci Sequence is...

7. Let n ∈ N with n > 1 and let P be the set of...

7. Let n ∈ N with n > 1 and let P be the set of polynomials with coefficients in R. (a) We define a relation, T, on P as follows: Let f, g ∈ P. Then we say f T g if f −g = c for some c ∈ R. Show that T is an equivalence relation on P. (b) Let R be the set of equivalence classes of P and let F : R → P be...

every analytic function is locally 1-1 whenever its derivative is nonzero): Let Ω⊂ℂΩ⊂C be open, and...

every analytic function is locally 1-1 whenever its derivative is nonzero): Let Ω⊂ℂΩ⊂C be open, and let ?:Ω→ℂf:Ω→C be 1-1 and analytic on Ω Then ?′(?0)≠0f′(z0)≠0 for every ?0∈Ωz0∈Ω. by contray suppose f'(z)=0

1. a) Prove that if n is an odd number then 3n + 1is an even...

1. a) Prove that if n is an odd number then 3n + 1is an even number. Use direct proof. b) Prove that if n is an odd number then n^2+ 3 is divisible by 4. Use direct proof. 2. a) Prove that sum of an even number and an odd number is an odd number. Use direct proof. b) Prove that product of two rational numbers is a rational number. Use direct proof. 3. a) Prove that if n2is...

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