3. Let R be the subring (a+b√10 | a,b e Z) of the field of real numbers. (a) The map N:R-Z given by a+b√10(a+b√10)(a - bic =a² - 10b² is such that N(uv) = N(u)N(v) for all u,v e R and N(u) = 0 if ar only if u = 0. ✓(b) u is a unit in R if and only if N(u) ±1.

Q: Let F be a field and let f(x) = a,x" + a„-p"-1 + · .. Prove that x - 1 is a factor of f(x) if and…

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Q: T: M2x2 (R) → R2 [x] is given by r(; ;) a С 3 (а +ь+с+d) + (а+ 2b + 3с + 4d)ӕ + (а + 36 + 5с + 7d)…

A: To solve this problem, we use the property  of linear transformation and to check condition which is…

Q: let k. be the splitting field of fax) &F (20) and let poxy be con irreducible factor of few in F(x).…

A: Let K be the splitting field of f(x)∈F(x) and let p(x) be an irreducible factor of  f(x) in F(x). If…

Q: Let phi / Q * (sqrt(2)) -> phi(sqrt(2)) be the map given t phi(a+b sqrt 2 )=a-b sqrt 2 . Show that…

A:

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: √2: s, 1€ Q}. Show that K satisfies the following: K, then x₁ + x₂ € K and X₁ X₂ € K. 9. Let K:= {s…

A: Given that, Now K satisfies the following:

Q: 1. Let T be a group of all invertible 2 x2 matrices of the form Where a.b.C ER and ac+0. Let / be…

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Q: 1. Based on the axioms of a vector space V over a field F prove the following: (a) If x, y z ¤ V and…

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Q: Let F be a field and aeF be such that [F (a): F]=5. Show that F(a)= F(x³).

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Q: (b) Q(V3). Show that the field of congruence classes Q[r]/(x² – 3) is isomorphic to

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Q: Let R be a U.F.D. and F its field of quotients. Then fo (x) any f(x) e F [x] can be written as f(x)…

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Q: Consider the number field F = Q(y), where y = /2+ v3. Find the irreducible polynomial f(x) of y over…

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Q: Let F be a field of characteristic 2 with more than two elements.Show that (x + y)3 ≠ x3 + y3 for…

A: Let F be a field of characteristic 2 with more than two elements, and let x, y be arbitrary elements…

Q: Let f(x) € Q[x] \ Q and KC C be the splitting field of

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Q: Let T: R² R² be given by T(x1, x₂) = -[at] It is known that I is an isomorphism. T-¹ is given by the…

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Q: a We have shown that C = b a the isomorphism f: CC where f - | a, b € a, beR is field. Show that CC…

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Q: a b 2.3 Let e: (M,(R), +) → be defined by e a + d. where a,b,c,d ER (a) Prove that e is a…

A: Given that θ:M2ℝ,+→ℝ,+ is a mapping defined by θabcd=a+d where a, b, c, d∈ℝ. We have to prove that θ…

Q: Let a ER which expressions are correct? and give the justification why they are part of the Borel…

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Q: We denote the set of polynomials with real coefficients as R[X]:= {ao + a₁X + a₂X² +.... + a₂X": n€…

A: (.)  Given set of polynomials,and linear map  defined as,    We will show that  is a linear operator…

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

A: Field of quotient

Q: H.w Pind the ro.t when [!,3} by Disection?

A: We have to find the root using Bisection method.

Q: 2.3 Let 9: (M2(R), +)→ be defined by e c d a b = a +d. where a,b,c,d €R (a) Prove that e is a…

A: A mapping f:(G,+)→(H,+) is said to be group homomorphism if it satisfies: f(a+b)=f(a)+f(b)…

Q: 2. Let R[x] be a ring over field R and let f, g are elements of R[x]. f=x3 +x2 +x +[0] , g=x +[1].…

A: We are given : f(x)=x3+x2+x+0⇒f(x)=x3+x2+xand g(x)=x+[1]⇒g(x)=x+1Now, Dividing f(x) by  g(x), we…

Q: ✓1.20 Show that these are homomorphisms. Are they inverse to each other? (a) d/dx: P3 → P₂ given by…

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Q: Which of the following are homomorphism? Monomorphism? Epiomorphism? Isomorphism? 1/ f: (Z, +)…

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

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Q: a. Show that the field Q(vZ. v3) = (a+byZ +cv3+ dvZ3: a, b,c, d e Q) is a finite %3D extension of Q.…

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Q: Exactly one of the following pairs consists of two vector spaces over the field R that are not…

A: Step 1: Step 2: Step 3: Step 4:

Q: Check whether or not if L/K and K/F are Galois extensions, then L/F is a Galois extension.

A: Suppose that EF and KF are extensions of a common base field F. Say that a field homomorphism φ: E→K…

Q: 2. Let R = F[X] and define three different R-module structures on F² (call them V1, V2 and V3), with…

A: Given R= F[X]

Q: Let  F=C  be the field of complex numbers and let V be any space over C . Suppose that f and g are…

A: I have provided solution in step2