3. In the field (Z7, +7,.7), then 7.7 (3-1 + 2-1) = (a) 0 (b) 1 (c) 5 (d) 6 (a) O (b) O (c) O (d) O 4. The field of quotients of (Z2, +2, -2) is (а) Qг (b) Z2 (c) Z4

Q: Find the smallest field that has exactly 6 subfields.

A:

Q: 32. Suppose B1, B2 are o-fields of subsets of 2 such that B1 c B2 and B2 is countably generated.…

A:

Q: . Show that x2 + 3 and x2 + x + 1 over Q have same splitting field.

A:

Q: Suppose that for polynomials J1/ 2J 3 over a field F the polynomial f is irreducible. Which is true…

A: Option {A} is correct. If f is irreducible  Than Statement ( A ) is true. Statement 3 and 4 not…

Q: What are the possible orders of the elements of the finite field F25 of 25 elements ?

A:

Q: Suppose f(x, y) = x² + 2æy+ y². Which field plot is Vf? Justify your answer. 鹿

A:

Q: Show that Q(√7, i) is the splitting field for x4 - 6x2 - 7.

A:

Q: Consider ζn = exp(2π√−1/n) ∈ C for n ∈ N. GIVE a diagram consisting of all subfields of Q(ζ6)…

A:

Q: 8.34. Let p be a prime and F a field with p² elements. Show that F cannot have more than one proper…

A: Given that F is a Field with q=p2 elements. It is required to prove that F cannot have more than one…

Q: In Z3×U3, Find the order of (1, 2). Show your work.

A:

Q: 19. Is Q[x]/(x² – 6x + 6) a field? Why?

A:

Q: 26 a ba Q(1E) = }a+bV2 / a, be Q4 2.1) Show (K,t,') eld under add. and rmult, of amotmias 2.2) Show…

A:

Q: (d) Z[ /2] is a field.

A: As you asked for option D.

Q: Show that Z3[x]/<x2 +x + 1> is not a field.

A:

Q: In Z[i], show that 3 is irreducible but 2 and 5 are not

A:

Q: 1. Suppose that (R x R, +,.) is a field where (x, y) + (z, w) = (x + z, y + w) and (x, y). (z, w) =…

A: Suppose that ℝ×ℝ, +, · is a field where (x, y) + (z, w) = (x+z, y+w) and x, y·z, w = xz-yw, xw+yz…

Q: Show that (AB)¹ = BTA¹ if A = 2 0 2 0 and B = -1 1 15) -3 1 -1 -1 2 2). a

A:

Q: 2 5= {(0, 0₂, 93) | DER₁, A₂ER₂, az ER₂ Z Show that sis subring of RxP₂Rz

A: We have to show that S is a subring of ring R1×R2×R3.

Q: 4: verify that wy = Wyx . ریاضيات المرحلة الثانية a w = In (2r + 3y) %3D

A:

Q: Show that Q(√7, i) is the splitting field for x4 - 6x2 -7.

A: x4 - 6x2 -7 = x4 - 7x2 + x2 -7                   = x2( x2 - 7) + 1 ( x2 - 7 )                   =…

Q: {0, A, Aº, $2} and X2 = {¢, B, Bº, S2} are o-fields, but &¡ U X2 is not. Why?

A:

Q: Determine the degree of the splitting field of x4 3 over Q. Justify your - answer.

A:

Q: find z trans foyms of folleing -an simwn ĭ

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Q: Find the splitting field of x4 + 1 over Q.

A:

Q: 3. Prove that Re(2-+z')<4 when z<1.

A: a.          x+y ≤x + y                                             (triangle inequality)b. Re(z1+z2)…

Q: 4. The field of quotients of (Z2, +2)-2) is (a) Q2 (b) Z2 (c) Z4 (d) Zg

A: Field of quotient

Q: Consider the field Z3[x]/. Please explain to me how this field has 9 elements. Thank you!

A: Consider the field Z3[x]/&lt;x2 +x+2&gt;. Please explain to me how this field has 9 elements.

Q: _. . Prove that z² = + Emo(-1)" (n + 1) (²2)", where |z-21 < 2.

A:

Q: By using Eisenstein's criterion we can conclude that the polynomials x^3+4x^2+3 and…

A: Using Eisenstein's criterion we can check a polynomial is reducible or irreducible. For that some…

Q: 18. Is Q[x]/(x² – 5x + 6) a field? Why? 10

A:

Q: Fiven with z1 = (-3,1) and z2 = (1, 4). Show that |21 2| = |21||2|- %3D

A: Complex number and it's modulus

Q: Show that x4 + x + 1 over Z2 does not have any multiple zeros inany extension field of Z2.

A:

Q: Q2: Show that whether (Z16,.0) is field or not?

A:

Q: The polyromial X+X+1 is ireducilole aver Z2. Use it to Canstruct additlon and multiplcotion table of…

A: Let F8 be a finite field with 8 elements. Since the polynomial x3+x+1 is irreducible in ℤ2x,…

Q: Find all subfield of Z14

A: Z14 is not a  field reas is given on the next step.

Q: Specify three properties that are special about conservative fields. How can you tell when a field…

A: The three special properties about the conservative field is given below:

Q: 1. Suppose that (R x R, +,.) is a field where (x, y) + (z, w) = (x +z, y + w) and (x, y). (z, w) =…

A:

Q: Given two polynomials in from of x^3+x^2+x+c =0 x^3+x^2+x+c =0

A: Given: To find their intersection using matrices. Given two polynomials in from of…

Q: How can I find all monic divisors of degree 1 in Z_7 [x], 5x^2+5x+4? please help me get started with…

A: Set of all Polynomials over a field ℤ7 is given by ℤ7x=anxn+an-1xn-1+ . . . +a1x+a0  where a0,a1, .…

Q: Prove or disprove that {Z109, +, x} is a Galois Field?

A:

Q: 1. Suppose that (Q x Q, +,.) is a field where (x, y) + (z, w) = (x + z,y + w) and (x, y). (z, w) =…

A:

Q: 5. Show that for Chebyshev polynomials (Tn, Tm) = 0, n #m, π, EIN n=m= 0, n = m = 0.

A: We have to solve given problems;

Q: 1. Suppose that (R × R, +, .) is a field where (x, y) + (z, w) = (x + z, y + w) and (x, y). (z, w) =…

A: I have proved (1, 0) is the identity element and the checked by option one by one

Q: Is the factor ring Z5[x]/ a field?

A:

Q: Prove that le-2z| 0.

A:

Q: Show that x^2 + x + 1 has a zero in some extension field of Z_2 that is a simple extension.

A: To show that the polynomial x2 +x+1 has a zero in some extension of Z2 :First let if possible, α be…

Q: [3 2 [1 4] 2) Consider the field F5. Let W = span C Max2(F5). Is [1 2] 1 e W? Your answer to this…

A:

Q: 1. Suppose that (R x R, +,.) is a field where (x, y) + (z, w) = (x + z, y + w) and (x, y). (z, w) =…

A: To find the inverse element of (1, 2) first we find the identity element in the given field, and…

Q: 3. Prove that the sandwitch theorem.

A:

Q: 7. Suppose that f is the scalar field that turns each point (x1, X2, X3, ... , x100) in R100 into…

A: Given that f: ℝ100→ℝ, defined by, fx1, x2, ……,x100 =x1+ x2+⋯+x100 (a) Then, ∇fx1, x2, ……,x100 =∂x1+…