9. Let Zs is a ring of integers mod 5 and f(x)eZ,(x) such that f(x) = x²+x+1, then: Zs a) f(x) is a field. b) f(x) has only two roots. c) f(x) has more than two roots. d) No one of the above. 55 4

Q: i) ∀a,b ∈ K, [0 c

A: This is a question of Logic.

Q: What is the field extension of Z[i]? Z[x]? explain reasoning

A: Field extension of Z[i] is  Q[i]  Since, the smallest field that contains Z is Q and Z[i]={a+bi | a,…

Q: Que 13: Each elements of a finite field F with p" elements satisfie the equation XP = x.

A:

Q: Let a be a root of f(z) = r – 2 and i be a root of z?+1. Let K = Q(a) and L= Q(i). Let F be the…

A: Let F be the composite field of K and L. Now, we need to find the degree of F over Q.

Q: 3) Show that (a + bi)º = (a – bi). Hint: for any x, y in a field containing Fp, (x+y)º = xº + yP.…

A: As per our guidelines, we are supposed to answer only one question if there are multiple questions…

Q: 2. Find all of the prime quadratic polynomials in F4[x], where F4 {0,1, a, b} be the field of four…

A: Given: F4={0,1,a,b}

Q: Read through the following theorem and proof and answer the questions on it that follow. Be sure to…

A:

Q: 4. Let R be a ring. (a) Let x, y, z € R. Prove that, if x | y and y | z, then x | z. (b) Prove that…

A: Let R be  a  ring .4) a)  Let  Suppose that   divides y and  y  divides z.We  have  to show that x…

Q: 12) Consider he algebraic extension E = bQ CVE, B, VF) of The field Q of rahionat numbers. Then (E :…

A: Given E=Q(2,3,5) of the field Q of rational numbers.

Q: 2. Which of the following is not an ordered field?

A: We have to find out which of the following is not an ordered field.

Q: Let R= Z/3Z, the integers mod 3. The ring of Gaussian integers mod 3 is defined by R[i] = {a+ bi :…

A:

Q: et FCK be an algebraic field extension and let a K. Prove that if dimp(F(a)) is odd then F(a) F(a²).

A:

Q: 12. Let K be a field. Define R с :) CEK}. : a, b, c E K a 0 b 0 a 00 a (i) Prove R is a commutative…

A:

Q: 14. Prove that F Z[]/(1+ 2i) is a field. How many elements are in F? What is the characteristic of…

A: Z[I]\(1+2i) and the number of elements

Q: (b) How many elements are there in S? Justify your answer.

A: The elements areNote: the elements of  are

Q: .55. (+) Let F be a field consisting of exactly three elements 0, 1, x. Prove that +x= 1 and that x…

A:

Q: (7) Let F be a field. Prove that (−1) · x = -X.

A: Let F be a field. We have to prove that (-1)·x=-x.

Q: 15. If S1 and S2 are two semialgebras of subsets of 2, show that the class S1S2 := {A1A2 : A1 € S1,…

A:

Q: 30. Let B; be o-fields of subsets of 2 for i = 1, 2. Show that the o-field B1 VB2 defined to be the…

A:

Q: 5. Let Q(r) be the field of rational functions over Q. Prove that Q(x)/Q(x²) is Galois, but…

A:

Q: (6) Let F be a field and x € F. Prove that -(-x) = = X.

A: Solution: Given that

Q: 14. Consider the quotient ring R = GF₁[x] / where me GF₁. For any specific value of m, the number of…

A:

Q: eed the ans

A: Given, Β is a σ-field of subfield of Ω and suppose Q : Ω↦0, 1 is a set function satisfying(a) Q is…

Q: Que 196: Let Q be the field of are rational numbers and ƒ(x) = x² + x² + 1 € Q [x]. Show that -1 +…

A: Let Q be the field of are rational numbers and  then we have toShow that…

Q: 1. Let a and b be elements of a field F. Show that if ab = 0 then either a = 0 or b = 0.

A: By Bartleby policy I have to solve only first one as these are all unrelated problems.These are all…

Q: 5.Let F be a field of char(F)=2. Then the number of elements xe F such that x = x is infinite

A:

Q: (8) If F is a field, then it has no proper ideal. От F

A:

Q: What is the order of the splitting field of x³ + x² + 1 = (x² + x + 1) ⋅ (x³ + x + 1) over Z₂?

A:

Q: Let α be a zero of x3 + x2 + 1 in some extension field of Z2. Solve the equation (α + 1) x + α = α2…

A: Given that α be a zero of x3+x2+1 in some extension field of ℤ2. We have to solve the equation…

Q: 3. Let F be a field and let a(x), b(x), c(x), d(x), and e(x) be polynomials in F[x] with a(x) ‡ 0F.…

A: To show that e(x) divides a(x)d(x), we need to prove that there exists a polynomial q(x) such that…

Q: Let S = Z[X] (x³ + x + 1) How many elements does S have? Justify your answer. Is S a field ? Justify…

A:

Q: Let F be a field. Prove that Fl) E F.

A: