Recall that a∈Z/nZ is called a primitive root modulon, if the order of a in Z/nZ...

Recall that a∈Z/nZ is called a primitive root modulon, if the order of a in Z/nZ is equal to φ(n). We have seen in class that, if p is a prime, then we can always find primitive roots modulop.Find all elements of (Z/11Z)∗ that are primitive roots modulo 11.

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Colby Messinger answered 2 years ago

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...

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...

In this problem, we will implement an nth root finder. Recall that the nth root of...

In this problem, we will implement an nth root finder. Recall that the nth root of x, written n√ x, is the number when raised to the power n gives x. In particular, please fill in the findNthRoot(int number, int n, int precision) method in the class NthRootFinder. The method should return a string representing the nth root of number, rounded to the nearest precision decimal places. If your answer is exact, you should fill in the answer with decimal...

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

Prove (Z/mZ)/(nZ/mZ) is isomorphic to Z/nZ where n and m are integers greater than 1 and...

Prove (Z/mZ)/(nZ/mZ) is isomorphic to Z/nZ where n and m are integers greater than 1 and n divides m.

If (x,y,z) is a primitive Pythagorean triple, prove that z= 4k+1

7. (16 pts) a. Show that 11 is a primitive root of 13. b. What is...

7. (16 pts) a. Show that 11 is a primitive root of 13. b. What is the discrete logarithm of 4 base 11 (with prime modulus 13)?

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.

(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...

Explain in simple terms what a primitive root is for prime modulus, and show that 2...

Explain in simple terms what a primitive root is for prime modulus, and show that 2 is a primitive root of 11, but 3 is not.

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