9. Let F be a field of characteristic 2 with at least 2 elements. Then for some x, y in F (x + y)³ ‡ x³ + y³.

Q: Suppose L: R² R³ is defined by L the given bases. [L(u)]s, = [L(u)]B₂ = ~Q[*^~-{]}]}----(HH:)…

A:

Q: Find the weighted Euclidean distance between the given vectors. = n with weights w = d(u, v) = Ex: 1

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Q: 4) Let F be a field. The derivation f' (T) € F[T] of a polynomial f(T) E a; · Ti e F[T] is defined…

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

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Q: Let F be a field. Show that if det(A)=0 with A E Mn(F) then A is a zero-divisor

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Q: Let α =21/8 and let K = Q(α,i). Show that (a) K is the splitting field of x8 −2 over Q and dimQ(K) =…

A: Given: α=218 and K=Qα,i.   To show: K is a splitting field of x8-2 over ℚ and dimℚK=16

Q: 13. If R = {a + b/2|a, bE Z}, then the system (R, +, ') is an integral domain, but not a field.…

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Q: Find the divergence of the field. F = (-2x+2y+3z)i + (8x - 2y - 5z)j + (6x + 8y - 7z)k div F =

A: Divergence of the field F is  ∇.F=∂∂xi+∂∂yj+∂∂zk.F

Q: 2. Consider the Newton polynomial Pn(z)= ao+a1 (1-10) + a₂(1-10) (1-11)+... +an(1-10)(1-1₁).…

A: Given that   that interpolate then we have Now  ,       putting

Q: 2.1. Suppose that F is a finite field with say |F| = pm that V has finite dimension n over F. Then…

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Q: Show that f(x)=x\power{6}-2ax\power{3}+a  is inseparable  over Z\index{3} (a), with a in any…

A: Given function is                             f(x) = x6-2ax3+a We have to show that this function is…

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: (1) Let R be an integral domain such that implies that (an) = (an+1) Show that R is a field. (ao) 2…

A: We have to show that integral domain is a field. A field is a ring in which every element is…

Q: Remark 2: Note that there can be several fields of order p". However, they will all be isomorphic,…

A:

Q: 25. Suppose 2 = {e'2T,0 < 0 < 1} is the unit circle. Let A be the collection of arcs on the unit…

A: We are given the set,  Ω={ei 2πθ, 0≤θ&lt;1}. which is a unit circle. Also, we are given a collection…

Q: Prove that 2/27/a is a field where p(X) = X³ + X + 1 € Z/2z[X] and a = = (p(X)).

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Q: Let A = 1 0 2 2 0 1 1 20 01 2 1002_ (a) Find the characteristic polynomial p(x) working over the…

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Q: 36. (Remainder Theorem) Let f(x) = F[x] where F is a field, and let a € F. Show that the remainder…

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Q: Determine if the following polynomials are irreducible or not k(1, y) = 1® – y° € Q!r, y]

A: Given:- k(x ,y)=x6-x7 ∈ Q x,y

Q: If o(t) is the real valued characteristic function of a r.v. X, then prove that : (ii) 1 + $(20) ≥ 2…

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Q: (a) (¹ (b) (₂ K) Define "Left Field" for a Relation R and a Function F. True or False and WHY? (No…

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Q: 27. Prove in detail that Q(/3+ 7) = Q(/3, /7). 28. Generalizing Exercise 27, show that if a+ b # 0,…

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Q: Problem 2. (a) Verify that the set Q(V3) = {r+sv3 |r,s € Q} is a subfield of the real numbers R.

A: By Bartleby policy I have to solve only first one as these 2 are unrelated and lengthy…

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

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Q: 4. Is M₂ (R) a field? Justify.

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Q: 1. Prove the following property in an order field F. If r + y 0, then r 0 and y= 0.

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Q: F(SuT)=F1(T) , where F1=F(S)

A: Given that                       F be field extension and S ,T are subset of K

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: Suppose f(x) E Z,[r] and f(x) is irreducible over Z,, where p is prime. (a) If deg f(x) = n, prove…

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Q: Let F2(t) denote the field of rational functions in t over F2. (a) Prove that F2(t)/F₂(t) is not…

A: Given Information: denote the field of rational functions in  over .To show:a)  is not Galois.b)  is…

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: 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: 3. Suppose F is a splitting field of æ" – 1 over Z3. (a) Find |F| if n = 3. (b) Find |F| ifn = 13.…

A: Given polynomial is  Let p(x)=x^n - 1

Q: 8. Let F be a field and let f(x) be an irreducible polynomial in F[x] Then if the characteristic of…

A: Let f(x) be an irreducible polynomial in Fx. The characteristic of F is zero. To show: f(x) has no…

Q: 1. Let F be the field with 2 elements. The number of (ordered) bases of F" over F is (а) п (b) п?…

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

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Q: Let F be a field and let f(x), g(x), n(x). k(x), and h(x) be polynomials in F[x]. Prove that if f(x)…

A: Given thatLet F be a field and let f(x) , g(x) , n(x) , k(x) , and h(x) be polynomials in F[x] .We…

Q: 1. Let F := 2/2z in each item below an element a and a polynomial f(X) € F[X] satisfyingin f(a) = 0…

A: The given data in this question is: The field F is the field with two elements, denoted by F:=ℤ/2ℤ.…

Q: 18. If F is a field and I is the ideal (x,xy) in F(x,y), then there are at least three dis- tinct…

A: Given: Let's consider F be the field and I be the x2,xy in Fx,y, then the distinct reduce primary…