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

(d) Use Lagrange's Theorem to prove that if n does not divide p-1, then F_p contains no nth roots of unity. [In fact, the converse is true: If n divides p-1, then F_p contains a (primitive) pth root of unity. We will prove this later.]

Expert Solution

Colby Messinger answered 1 year ago

(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...

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...

Prove: Every root field over F is the root field of some irreducible polynomial over F....

Prove: Every root field over F is the root field of some irreducible polynomial over F. (Hint: Use part 6 and Theorem 2.)

A field is a commutative ring with unity in which every nonzero element is a unit....

A field is a commutative ring with unity in which every nonzero element is a unit. Question: Show that Z_5 under addition and multiplication mod 5 is a field. (state the operations, identities, inverses)

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.

In this assignment, we will implement a square root approximater and then an nth root approximater....

In this assignment, we will implement a square root approximater and then an nth root approximater. Recall that the nth root of x is the number when raised to the power n gives x. We learned in class that we can use the concepts of binary search to approx imate the square root of a number, and we can continue this logic to approximate the nth square root. Please look at Lecture Notes 03, section 4 for a little more...

Prove that n, nth roots of unity form a group under multiplication.

In C++ In this assignment, we will implement a square root approximater and then an nth...

In C++ In this assignment, we will implement a square root approximater and then an nth root approximater. Recall that the nth root of x is the number when raised to the power n gives x. We can use the concepts of binary search to approximate the square root of a number, and we can continue this logic to approximate the nth square root. We're going to use the concepts of binary search to try and approximate ending the square...

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

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)?

Question