(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 1 year 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

. Let xj , j = 1, . . . n be n distinct values. Let...

. Let xj , j = 1, . . . n be n distinct values. Let yj be any n values. Let p(x) = c1 + c2x + c3x 2 + · · · + cn x ^n−1 be the unique polynomial that interpolates the data (xj , yj ), j = 1, . . . , n (Vandermonde approach). (a) Remember that (xj , yj ), j = 1, . . . , n are given. Derive the n...

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