Let f (x) belong to Z[x]. If a mod m = b mod m, prove that f (a) modm = f (b) mod m. Prove that if both f(0) and f(1) are odd, thenf has no zero in Z.

Q: Let R= Z/5Z, the integers mod 5. The ring of Gaussian integers mod 5 is defined by R[i] = {a+ bi :…

A: Def zero DevisorIn a ring R , an element a∈R will be zero devisor if there exists a non-zero b∈R…

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

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Q: Let n > 2 be an integer. Show that Z/Zn is a field if and only if n is a prime number.

A: Let A=1¯,2¯,3¯,...,n-1¯ we show for ⇒ Let ℤℤn is a field⇒∀a∈A,∃x∈A such that ax=1¯ax≡1modn has a…

Q: Let R= Z/5Z, the integers mod 5. The ring of Gaussian integers mod 5 is defined by R[i] = {a+ bi :…

A: A non-zero commutative ring with unity having atleast two elements and without zero divisor is…

Q: Let m be a positive integer. Show that a mod m = b mod m if a ≡ b (mod m)

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Q: Show that if E is a finite extension of a field F and [E : F]is a prime number, then E is a simple…

A: Let, α∈E be such that α∉F. As we know that, If E is the finite extension field F and K is finite…

Q: The notation x = y mod n is used to indicate the unique integer x in the range 0 :$ x < n such…

A: Given statement is that the notation x = y mod n is used to indicate the unique integer x in the…

Q: Suppose a E Z. Prove that if a³ 1 (mod 5), then a # 1 (mod 5).

A: In this question, we use the method of contradiction to prove the desired result.

Q: 2. Let a be a positive integer. Explain why the linear congruence ax = b (mod a + 1) has a solution…

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Q: For any integer a, prove that a = 0 or 1 (mod 3).

A: Every number is either of the forms 3k, 3k−1, 3k+1 where k is an integer

Q: Suppose that E is an extension of F and a, b E E. If a is algebraic over F of degree m, and b is…

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Q: For integer n>1, nZ={nx:x in z}, nZ is set of all integer multiples of n. Prove that integer p is…

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Q: Prove that for every field F, there are infinitely many irreducibleelements in F[x] .

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Q: (A) In F;[x], find a complete set R of the least degree representatives modulo m(x) = (4x + 3x+ 3).…

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Q: 1. Let X = {z €R:r#0,1} and define f : X X by f(r) = 1-. Prove or disprove that f is a permutation…

A: As per our guidelines we are supposed to answer only first one. kindly reposed other part in the…

Q: Let r be a primitive root of some n > 3. Prove that r = -1 (mod n).

A: Solution: Let n be any positive integer. Let Un= r : r is less than n and prime to n , where r=r+kn…

Q: For any integer a, prove that a=0 or 1 (mod 3).

A: The solution is given as follows

Q: . Show that Z with addition modulo m, where m≥ 2 is an integer, satisfies the closure, associative,…

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Q: 2. Let a be a positive integer. Explain why the linear congruence az b (mod a + 1) has a solution in…

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Q: Prove that the number of solutions to the congruence xk ≡ 1 mod p is gcd(k, p − 1)

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Q: Let a be a quadratic residue of the odd prime p. (1) Prove that a is not a primitive root modulo p…

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Q: Let p be a prime number. The polynomial p(x) = x²+x+1 is irreducible over Zp if and and only if…

A: Given that for a prime number p, the polynomial px = x2+x+1 is irreducible over ℤp if and only if…

Q: Suppose that p is a prime such that p = 5 (mod 8). Let k E Z such that p = 8k+5. For any a E Zp,…

A: 1. Let a∈ℝn. Then a is called quadratic residue modulo n if there exists an integer x such that…

Q: Let (an) be a sequence of real numbers that diverges to +∞, and let (bn) be bounded. Prove that lim…

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Q: Let p = 2^k + 1 be a prime for some k ≥ 1. Let a be any integer such that gcd(a, p) = 1 and a is NOT…

A: Consider the given information:

Q: The first term in the Maclaurin's expansion of f(x)= In(1 – 2x) is O-2 o not defined 0.

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Q: Let F, denote the nth Fibonacci number (F1 = F2 = 1, Fn+2 Fn+1 + Fn for n > 1). Use %3D induction to…

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Q: 3. Suppose f(x) and g(x) are monic integer polynomials. Prove that f(x)|g(x) in Q[x] if and only if…

A: Integer polynomials are polynomials having integer coefficients. And the monic polynomials have…

Q: s) Let n 2 2. The ring Z, of integers modn is called a field if and only thbe only solutions to the…

A: The congruence ax≡bmodn is solvable and its solution is uniquely determined modulo n. ℤn is a field…

Q: Find all positive primes p such that n = 1001p satisfies that a" = a mod n for all a.

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Q: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

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Q: 6. Let Fn 22" + 1, n > 1, be a prime Fermat number. Show that 3 is a primitive root modulo Fn.

A: Given: Fn=22n+1,n>1

Q: Prove that xk = x* mod |G| for x € G for any integer k.

A: Prove it considering two cases.

Q: 1. Use Euler's Theorem to prove a 265 = a (mod 105) for all a E Z.

A: To prove:                   a265≡amod 105 for all a∈ℤ by using Euler's theorem.   Euler's theorem:…

Q: .3. Let K be an extension of a field F. Let

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Q: *).). (a) Show that X is measurable with respect to o-field G if and only if o(X) C G. Show that X…

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Q: 1. Use Euler's Theorem to prove Q265 = a for all a E Z. a (mod 105)

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Find the Maclaurin expansion of In (cos(x)). О а. x2 x4 x6 17x8 + .. 2520 - 2 12 45 O b. x4 x6 17x8…

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Q: Find an integer x satisfying the two congruences: x = 1(mod 3) 2x = 0(mod 6), or show that no such…

A: The given congruences are                                 x≡1 (mod 3)2x≡0 (mod 6)

Q: Find the integral basis and discriminant of the number field K = Q(/=p), where p is a prime such…

A: It is given that

Q: 2. Suppose F is a finite field. Prove that F× is cyclic. Deduce that x² = -1 has a solution in a…

A: Here, F× denotes F-0. We need to prove F× is a cyclic group under multiplication.

Q: 1. Use Euler's Theorem to prove a = a (mod 105) for all a E Z.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 6. Let f(x) be a polynomial with integral coefficients (like f(x) = 3x^^5-x^2+7) and suppose there…

A: Given:- fx=3x5-x2+7 There are 2 solutions  x mod 5 such that fx is congruent to 3 mod 5 and there…

Q: Let A = {x e Z: (7x + 39) mod 17 = 14}. a) Find a number ß such that (7B) mod 17 = 1. b) Prove that…

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Q: K be a field extension of a fi is algebric over F if and only of 5 that [56)

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Q: Prove that the ring of polynomials with coefficients from P is under the ring of formal series…

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Q: 2. If K is an extension of F and [K : F] is a prime number there is no field L such that FCLCK.

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Q: Let F be a eld of characteristic p > 0. Let a EF be such that ais not a pth power of any element of…

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Q: Prove that for any positive integer n, a field F can have at most afinitenumber of elements of…

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Q: Show that P is irreducible in Q[x].

A: To prove: P is irreducible in Q[x]