Let IC Q[x] denote the principal ideal generated by x – 26x + 52. Prove that E := Q[x]/I is a field. Give an example of two cosets p(x)+I and q(x)+I with deg p > 2 and deg q > 2 such that (p(x) + I)(q(x)+I) = 1+I. Give a basis for E over Q.

Q: 3. Let p₁(x) = ½x(x − 1), p2(x) = −x² + 1 and p3(x) = ½x(x+1). - (a) Show that E = {P1, P2, P3} is a…

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Q: 8. Consider the basis * B = {1 + x + x², x + x², x?}. for P2. Let p(x) = 2 + x+ 2r². Then [p(x)]8 =…

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Q: Let T: R* → P2 be defined as T | | = (a + b)t² + (c – d)t + d. a. Find T b. Find a basis for the…

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Q: Let V = P2(R), give V the inner product defined by (f,g) = Lo s(1)g(t) dt, and let W = span {1 – x}.…

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Q: Let T = L(V, W) where V and W are finite-dimensional vector spaces over the same field F. Let…

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Q: Show that the cubic field generated by a root of f(x) = 3x2 base. Use the fact that…

A: From the given information. The function f(x) is as follows.

Q: Consider the following polynomials in P • P4(x)=x+1; Ps(x)=²+1; • Pu(x)=2²³ +1. P₁(x) = x³ + x² +1;…

A: The given polynomials are p1x=x3+x2+1p2x=x3+xp3x=x2+xp4x=x+1p5x=x2+1p6x=x3+1 (a) We have to find a…

Q: Let f E P(R) be a polynomial of degree n. Let yo,..., Yn be the dual basis of 1, x, x²,..., x". Show…

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Q: 15. Consider the basis * B = (1+** ****). for P2. Let p(x) = 2 + 3x + 2x. Then [p(x)la = O None

A: Explanation of the answer is as follows

Q: Let f(t) = (t – 2)*(t+1)² be the characteristic polynomial for a linear operator T. Obtain all…

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Q: Consider P2 (R) together with inner product: Find an orthogonal basis for P2 (R). (p(z),9(2) = Г'р…

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Q: (a) Check whether or not x³ + 2x + 1 e F;[x] is a primitive polynomial. (b) Let F, L, K be fields…

A: 6. (a)  In a principal ideal domain R a polynomial is said to be primitive polynomial  if the…

Q: Let R be the field of real number and {e1, e2} be the standard basis for R2 . Consider the linear…

A: Let R be the field of real numbers and {e1,e2}  bet the standard basis. Consider the linear operator…

Q: Find the degree and a basis of the extension field Q(v3, v7) Q(V3 + v7). over

A: This question is from field theory's topic extension of field. Solution is given below.

Q: Let T be a linear operator on R³ such that the rational form of T is [T] B = 0-20 1 0 for some…

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Q: 1. Let a = √3+√3 (a) Find ma(z) F raational numbers). the minimal polynomial of a over Q (the field…

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Q: 3. Let E be the elliptic curve E(F3). y² = x³ + 2x + 1 defined over the field F3. Compute

A: The given elliptic curve is y2=x3+2x+1. To Find: EF3. As per policy, only first question is solved…

Q: Let T: P₁ → R² be defined by T (P) Let q(t) = 5t6t² +7t³. Find T(q). Find a basis for the kernel of…

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Q: .3 Let W = span{x, x' x³, x>} in C[0,1]. Use Gram Schmidt to find an orthogonal basis for W.

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Q: Let (X, |-||) be a normed space. Prove that if X has a Schauder basis, then X is separable. (Here,…

A: Schauder basis: If a normed space X,. contains a sequence en with the property that every x∈X, there…

Q: F,[x] is the vector space of all polynomials over a field F of degree at most n and has a standard…

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Q: Let V = Poly4(R) and let B = (1, x2 − 1, x, x^3 − 2x, x^4 − 3x^2 + 1). (a) Find a basis C of V such…

A: Let V = Poly4(R) and let B = {1, x2 − 1, x, x3 − 2x, x4 − 3x2 + 1}. Note that here V is a vector…

Q: Take L: V --> W, from P2 --> P2 and L(p) = p + p', with basis B = {1 + x + x2, 1 + x, 2x + x2} and…

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Q: For the number field K = Q(V7): (a) Find the minimal polynomial of V7 over Q. (b) Find all three…

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Q: Let 5 be a primitive 9th root of unity in C and consider the extension Q(S) of Q. What is [Q(5) :…

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Q: 9. Find an orthonormal basis for P[r]3 with a basis 1, 2, 2² if the inner product on P[r]3 is…

A: We have to find the orthonormal basis for  with a basis The inner product is defined by

Q: Consider the basis S = (v, v2) for R², where VI = (-2, 1) and v2 (1.3). and let T: R2 R' %3D be the…

A: A detailed solution is given in step 2.

Q: (i) Find gcd(2,3+5i) in Z[i], (ii) Show that 3 + 4i and 4-3i are associates in Z[i] (iii) Determine…

A: (i).

Q: Find the splitting field E of f(x)-4-5 over Q, determine [E: Q), and find a basis for E over Q.

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Q: 10. Let o: Q(V2) → Q(/2) be the map given by ø(a+bv2) = a-bv2. Show that is a Q-linear field…

A: We have given a map ,  ϕ : ℚ2 → ℚ2  such that , ϕa + b2 = a - b2 We need to show that , ϕ is an…

Q: 9. Use the field norm to show: a) 1+/2 is a unit in Z [/2] b) -1+v-3 is a unit in Z [1+-3 ]

A: Use the field norm to show thata1+2 is a unit in 2.b-1+-32 is a unit in -1+-32.

Q: Find an integral basis for the field Q(-5+ V-28)

A: We have to find the integral basis for the field ℚ(-5+-28). So here our field is given in the form…

Q: Let T: R² R² be defined by T Letu-de-{---G} = an and C= = V₁ = T(u) 3 - X1 X2 Ex: 5 , V2 = Find…

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Q: Let F = Q(π3). Find a basis for F(π) over F.

A: It is given that,  F = ℚπ3. We have to find a basis for Fπ over F. We know that π is a…

Q: Let the complex field C be a vector space over the real field R and the conjugation operator T on C…

A: The given mapping is given by:Tz=z¯We need to finda the matrix of T relative to the usual basis 1,ib…

Q: Let B be an n × n matrix over a field F, and suppose that B has minimal polynomial x2 +x +1. (i)…

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Q: Let F be a field and let f(x) E F[x] be a

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Q: The Fourier transform of f(t) is F(w) = 2e 6jw 9+ 3 Using the Convolution Theorem, find f(t) in the…

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Q: Let V = span {V1, V2}, where V1 and V2 = 1 2 Find a basis S for R3 that includes vi and v2.

A: Let V  = span { v1 , v2}  where   v1= 102 and v2 = 111 Find a basis S for R3 that includes v1 and…