What is the likely amount of solutions to the problem xt = a in the field F*p and does is this determined by a? Given that p is a prime number, t is any divisor of p-1, and an a that is an element in the field F*p.    I know the best way to solve this problem is by first setting a value r that is a primitive root (r (mod p)) and then taking the logr of x and a. Which can then let the original equation become t (logr(x)) ≡ (logr(a)) (mod p − 1) which will help calculate the possible amount of solutions for the new equation. I need someone to explicitly label the steps for me.

Q: Let p(x) be a polynomial with integer coefficients, and let n e N. Given two integers a and b such…

A:

Q: Question 6. The goal of this exercise is to make sure you understand √x² = ]x] and in general √x²…

A: a) The fact that for any real number a such that a>= 0, there is a unique number sqrt(a)>= 0…

Q: Question 2 2.1. Suppose that y = 22.7184. Round y off to four significand digits 12.5. What is the…

A:

Q: Show that if ya – ay = 1 has no rational solution for positive a such that a is a natural number and…

A: The given equation is ya-ay=1,a≠2 this equation is not valid for a = 1 although it is given for…

Q: Problem 2.9. Let 9 be a root of an irreducible polynomial p E Q[r]. Prove that if 0 € C, then deg…

A:

Q: (d) (x² + 2x + 1)q(x) = 1 (mod x³ + x² + 1) over Z3

A:

Q: i. Suppose p is an irreducible polynomial of degree n with coefficients in Q. Then p has

A: Determine whether p(x) = x3+x+1 is irreducible over  Sol: we know that , let f(x) be a polynomial in…

Q: Find phi(450), the Euler’s Phi function.  (Note that 1,2,3,5, 7, … etc. are the primes)

A: Note: We are entitled to solve only one question at a time. To  Find: phi450 Concept: (a). Euler's…

Q: Let p be a prime and let q = p². Show that Z,[x] has a unit that is not a constant polynomial.

A: Given that, Let p be a prime and let q = p ^ 2 . Show that Zq[x] has a unit that is not a constant…

Q: There exists an integer m > 3 such that m2 – 1 is prime. The above statement is due to proof by…

A: We have to complete the given sentence.

Q: What are the coefficients of æ5000 and æ4999 in the simplified expansion of (5æª – 2)²2021?

A:

Q: \2 ) **-8 =O x38

A:

Q: IF TE A (V) and if 9 (1) € Fox) is such that a (T) = 0, is a characteristic root it true that every…

A: The statement is false.

Q: 2. For a, b e R, show that r3+ ax +b= 0 has three real roots iff 4a3 + 2762 < 0. Remark: This is a…

A: We have to show that, x3 + ax +b =0 has three real roots iff 4a3 +27b2 &lt;0 for any real numbers a…

Q: Question 3 Let p be a prime number. We denote by fp(x) the polynomial x3 -9 regarded as an element…

A: Let fpx=x3-9 where p is a prime such that p-1 is not divisible by 3. Eisenstein criterion: If…

Q: Derive the following alternative form of Wallis’s product formula:

A:

Q: (c) Find integers p and q so that 17p + 45q = 1. Show your working steps. (d) Using the result in…

A: Given that 17p+45q=1 has no solution such that p and q are integers which means gcd of (17, 45) is 1…

Q: Find the order of the residue class 5 in Z/29Z.

A:

Q: The polynomial f(x) = x³ − x−3_mod (47) has x₁ = 10 as a root. Use Hensel's Lemma to lift this root…

A: Given Information:The polynomial  is given as .The one of the roots is given as .To find:The lifted…

Q: Theorem 1. Let r, y, z be positive integers where r is odd and y is even. Then r, y, z is a…

A: As per Bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: deal with Hermite’s equation: y′′−2xy′+2αy =0, −∞&lt; x &lt;∞.  ...........(11.3.13) Use some form…

A: note : Since you have posted multiple questions, we will provide the solution only to the first…

Q: 1. (i) Find all four square roots of 61 in Z247. (ii) Show that r³7 = x for all r E Z217- (iii) In…

A: (i)      Let 'a' be square root of 61 in Z247 Then,       a2   = 61 (mod 247)     i.e.,  a = 247 K…

Q: Suppose you have a value (g = 23) in the field F79. Without computing any powers of g or any…

A:

Q: Let g(x) 3 +x+7. = X i) Find a solution to g(x) in Z/21Z. ii) Find a solution to g(x) in Z/3969Z…

A: We will use the basic knowledge of polynomials and the elementary theory of the congruence modulo…

Q: . Let f (x) = 11x +9x +21. Prove that f (x) is irreducible over Q.

A: We solve this problem using  (mod p ) irreducibility test over f(x) where p = 2 , a prime number.

Q: 1.27. Without using the fact that every integer has a unique factorization into 1 and if a | bc,…

A:

Q: Use the following values and five-digit rounding arithmetic to construct the Hermite interpolating…

A:

Q: Find a primitive root in Fq = 137, which is not 6, say a . Show that a is indeed a primitive root.…

A: And,

Q: (2) Prove that (1+x)" > 1+nx for all x > 0 and n > 1.

A:

Q: 1 Use the Newton forward-difference formula to construct an interpolating polynomial of degree 3 to…

A:

Q: In this project, we use the Maclaurin polynomials for f(x) = e x to proved that e is irrational. The…

A: In this project, we need to use the Maclaurin polynomials for f(x) = ex to prove that e is…

Q: 2.3. Let g be a primitive root for Fp. (a) Suppose that x = a and x = b are both integer solutions…

A:

Q: 2.1 Suppose that y = 22.7184. Round y off to four significant digits. 2.2 What is the Lagrange's…

A:

Q: olve and answer the following problems as indicated. 2) Create the multiplication table for…

A:

Q: Suppose you have a value (g = 23) in the field F79. Is g a primitive root (do not compute all powers…

A:

Q: Define the following operation on Zn: S [1] if a = b (mod 5) (a] + [b] = { joj if a ±b (mod 5) %3D…

A: Consider the provided question, The provided operation on Zn is given by, a*b=1    if a≡b(mod5)0…

Q: Theorem 5.3. Let a be an odd integer. If a3 we have (29) a 2 = 1 (mod 2º), so there are no primitive…

A: Proof:                 We shall prove by induction on α. First we consider α=3. Let α be an odd…

Q: [1 + (- P] f S + (−1)²] XP 1+x² XP 1+x² dx = implifying and using the given formula for the pth…

A:

Q: Let 5 be a primitive 9th root of unity in C and consider the extension Q(5) of Q. Using the formula…

A:

Q: 3. (i). Suppose f (x) is a polynomial with integer coefficients such that f(2) = 0(mod 5) f(4) =…

A: Given that  (a) f(2)≡0(mod 5)f(4)≡0(mod 9) We have to find a polynomial such that f(x)≡0(mod 45)…

Q: Use the Chinese remainder theroem to find all of the soutions to x2+1=0, modulo 1313.

A: We will convert this congruence equation to a linear congruence system then we will use Chinese…