Consider the quotient ring Q[x]/< x² + x – 2 > (a) Give an example of a zero divisor in this ring. Give the multiplication which shows it is a zero divisor. (b) Prove that if a + bx + < x² + x- 2 >= c + dx + < x? +x – 2 >, then a = c and b = d (c) Determine (x + < x² + x – 2 >)-1 as follows: (i) Briefly explain why (x + < x² + x – 2 >)-1 = a + bx + < x² + x – 2 >, for some | a, b E Q (ii) Use part (b) to determine a and b (iii) Verify that your answer is in fact the inverse.

Q: Find the dimension of the splitting field of each of polynomials x+1 and x² - 2.

A:

Q: Find the inverse of X + (X² – 8X + 7) in the form aX + b in the ring Q[X]/(X² – 8X + 7). - a b =…

A:

Q: Generate the 2nd and 3rd degree Legendre polynomials.

A: Since we know the Rodrigue's formula of Legendre polynomial is Pn(x)=12nn!dndxnx2-1n For second…

Q: (x + 5)² = x² + 10x + 25

A:

Q: Decide whether z(sqrt(-5)) unique factorization domain or not (ring theory)

A: To decide whether Z(√-5) is unique factorization domain or not.

Q: 3) Show that (a + b +c)² = a² + b² + c² + 2ab + 2ac + 2bc is a polynomial identity.

A: To show: a+b+c2=a2+b2+c2+2ab+2ac+2bc

Q: In Q[r]/(x² - 4) give an example of a zero divisor. Justify.

A:

Q: Consider the system Ax = b where A = [11] and b= [¹₂] A basis for the null space of A is given by…

A:

Q: 17) Explain why these two quotients are different. Compute each one. (x – 2)* x* - 16 and X-2 x- 2

A: (a-b)2= (a-b)(a-b) and a2-b2=(a-b)(a+b)…

Q: Use the following matrices for this activity. [-2 0 3 3 1 2 0] -1 -2 3] 1 C = | 2 L-i 0 1 -1] A = B…

A:

Q: Is Q [x]/⟨x2 -5x + 6⟩ a field? Why?

A:

Q: Calculate the commutators [p(hat)x3, x(hat)].

A: We have to calculate the following commutators. p^x3 , x^ Introduction: Commutation of operator A^…

Q: Write x3 + 2x + 1 as a product of linear polynomials over someextension field of Z3.

A:

Q: Multiply the polynomials (x-1)(x* +x³ + down an identity for (x - 1)(x" +x"-1 +..+x² +x + 1) for n a…

A: Multiply the given polynomials using table (x-1)(x4+x3+x2+x+1)   x4 x3 x2 x 1 x x5 x4 x3 x2 x…

Q: f = x² + x³x²+1, g = x³ + 1 in Q[x];

A: Since you have posted multiple questions, we will solve the first question for you. If you want any…

Q: Show that the polynomial x + x

A:

Q: Use the definition of O-notation to prove that 15x3 + 8x + 4 is O(x3)

A:

Q: Write x3 + 2x + 1 as a product of linear polynomials over some extension field of Z3.

A: Given: x3+2x+1. To Do:We have to write x3+2x+1 as a product of linear polynomials over some…

Q: Check whether or not x³ + 2x + 1 € F5[x] is a primitive polynomial.

A: Primitive polynomial

Q: (4) Find a polynomial that has 1+i and 1- i as zeros.

A: We have to find polynomial

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

A:

Q: ísomorp

A: Isomorphic to quotient ring:

Q: Find the standard inner product on P of the given polynomials. p= -2+ 4x + a2, q = 3+ 4x 3a2. (p, q)

A:

Q: Write x3 + 3x² + 3x + 4 in Zs[x] as a product of irreducible polynomials.

A:

Q: polynomial in V2+i and compute the conjugates of i over Express i as a Q(/2 + i).

A: As per the question we have to express i as a polynomial in (√2 + i) and then compute its conjugate…

Q: Find the zero-divisors of Z6.

A:

Q: A. Classify the following polynomials as to degree. 1. 4x2 + 3x - 7x + 8 6. 3 + 3x? 2. x + 2x - 5 7.…

A:

Q: 2. Let A, B, C, and D be four groups such that A-C and B D. Show that A x B Cx D.

A: We will use the face that A isomorphic to C  and B isomorphic to D to prove the asked result.

Q: If the conjugate of a zero (which is 2+3i), would it still be 2-3i and would 2-3i count as one of…

A:

Q: Question: In the ring 7L8 [x]. Show that 1+2x is a unit. a Unit.

A: Definition: Suppose R be a ring with unity 1 and a non zero element a∈R, then we say that a is a…