For each the following, write TRUE if the statement is always true. Otherwise, write FALSE. No need to show solutions/proofs. (a) (i) ◄ Qs (b) char(Z₂ × Z7 × Z₁4) = 14 (c) Z₂ xZ7xZ12 Z14 × Z12 (d) A commutative ring R is a prime ideal of itself. (e) If p and q are primes, then there is a unique abelian group of order pq.

Q: The characteristics of local rings is either 0 or prime.

A: Let R be a local ring (with 1). Suppose char R = n ≠0. Let, if possible, n be divisible by two…

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Q: 29 justifying each step you use for this. 57 b) Calculate

A: (b) The objective is to find the value for 2957. By using the division algorithm, 29=0×57+29. So,…

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Q: Prove ân = √nn-1 and atyn = √n + 14n+1. Hint: use the recursion relations for Hermite polynomials.

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Q: Z3-22 If z1 = -4 - j5, Z2 = -3 + j2 , and z3 = 2 - j3, what is Re Note: z* is the complex conjugate…

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Q: Abstract Algebra. Please explain everything in detail.

A: To describe all the field automorphisms of the given field.

Q: (6) The ring (Z,12, +12,·12 ) is ...... (a) field (b) semi-simple (c) simple (d) No Choice O (a) O…

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Q: (a) Let R be a ring and OR E R be the zero element of R. Then check as to whether or not {OR} is a…

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Q: (10 +3i2-5)-(11 + 10i)

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Q: expr lim x→-8 -2x² - 20x - 32 x + 8 - 32)

A: Given problem:-

Q: Check if the solution is correct. Correct the mistakes. If x4=x for any x in R, prove that R is…

A: Given that x^4=x for any x in R. To prove R is commutative.

Q: Ex: tet R = {o, a,b,c} a G O a b ( وه هه 2 2 G 6 ( 0 a C b a O a b a b 2/2 ( в с @ Show that (B,…

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Q: Please use method from Abstract Algebra, Show that the polynomial 37a? + 30a + a4 = 1500 does not…

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Q: Let a and b be arbitrary elements of a ring R, whose characteristic is 2 and ab = ba. Then (a) (a +…

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Q: őf a ring R, e characteristic is 2 and ab = ba. Then コ+b)%=α'+b@ コーb?=α'+b? o!h (a) and (b) Ione of…

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Q: Q 10 Which one of the following is not a ring? (a) (Z, +, ·) (b) (M4x4(R), +, ) (c) (Zs,O5, 85) (d)…

A: We know that of G is group then G is follow closure property .  G is follow associative property.…

Q: 12. Which of the following polynomials is reducible over Q а) 4x3 + x — 2 b) Зх3 — 6х? +х - 2 c) 5x3…

A: Reducible, Irresistible

Q: 1. Find all units in Z[V-3]. Justify your answer. 2. Prove that 2 is an irreducible element in…

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Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

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Q: Which of the followi:ng statement is false? Ta) The ring (Q, +, .} is an integral domain. (b) The…

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Q: Let K be integer ring module 12 and let I=([4]) and J-([6]) be ideals of K. Then ([0])

A: Let * be integer ring module 12.  Let  I=4J=6

Q: If w and af are the nonreal cube roots of unity and [1/(a + c)] + [1/(b + w)] + [1/(c + w)] = 2w²…

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Q: (a) What is the degree of Q(7, V3) over Q ? Justify your answer. Is the polynomial X - 5 € QIX)…

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Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

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Q: The characteristic of the ring 6x8 ×; is 14 (A) 2 (B) 48 (C) 84 (D) 168

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A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: Which one of the following is true? In the ring Z6, 4 does not divide 2 In the ring Z6, if x² = x…

A: Notice the some points regarding ℤn : ℤn=0,1,2,3,⋯⋯⋯⋯⋯,n-1. If x,y∈ℤn and x divides y then there…

Q: Please use method from Abstract Algebra, Show that the polynomial 37a + 30a + a4 = 1500 does not…

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Q: (c) Let R be a finite commutative ring with identity. Prove that an ideal in R is prime if and only…

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Q: Give the smallest extension of Q that contains all the zeros of (x4 – 8x² + 15)(x² + 1). -

A: Yes

Q: The rings Z and 5Z are isomorphic.

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Q: Simplify the following expressions in the indicated polynomial rings: A. (3x³ + 2x² + 4x + 2) + (2x¹…

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Q: decide if the given statement is true or false,and give a brief justification for your answer.If…

A: As per the question we have to check whether each of the Legendre polynomials contains terms with…