Describe the group of the polynomial (x4 – 5x2 + 6) ∊ ℚ[x] over ℚ.

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

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Q: Let G=Z5xZ6X Z9 X Z12- Find the elementary divisors and invariant factors of G.

A: Solution :-

Q: Calculate GCD(x3 + 2x2 + 3x , x2 + 3x + 2) in Z5 [x].

A: We have to find GCD x3+2x2+3x,x2+3x+2 We will use Euclidean Algorithm Let…

Q: Find a monic polynomial of least degree in R[x] that has 1-i and 2i as roots. write the powers of x…

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Q: In Z3[x], show that the distinct polynomials x4 + x and x2 + xdeterminethe same function from Z3 to…

A: We know that ℤ3 is a field i.e., a commutative ring with unity in which every non-zero element has a…

Q: . Factor x12 -1 as a product of irreducible polynomials over Z.

A: Given: x12-1. We have to factor x12-1 as a product of irreducible polynomials over Z.

Q: In Z3[x], show that the distinct polynomials x4 + x and x2 + x determinethe same function from Z3 to…

A: Given: x4+x and x2+x are two distinct polynomials in ℤ3x.We need to show that the given two…

Q: Express the polynomial x5 + 3x3 + x2 + 2x as a product of its irreducible factors in Z5[x].

A: Given: [May be the question is wrong, otherwise the given solution is correct.] The given polynomial…

Q: Determine which of the polynomials below is (are) irreducible over Q.a. x5 + 9x4 + 12x2 + 6b. x4 + x…

A: According to Eisenstein's criterion, it provides a suitable condition for polynomial be irreducible…

Q: Determine which of the polynomials below are irreducible over Q. (a) a" + 9xª + 12:r² + 6 (b) a* + x…

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

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Q: Find the coefficient of x5 in (1 – 2x)-7 using generating function.

A: Given to find the coefficient of x5 in 1-2x-7 The generating function will be Gx=1-2x-7…

Q: The set of polynomials {1, x² + 3x“, 2x² – 7x4} is a linearly independent set in P4.

A: NOTE: Due to time limit answer of 1 is given here, for other please ask in a different question.…

Q: Is z(√2) isomorphic to z(√3)?

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Q: Let g(x) = 4x3 + 4x - 10x2x3 be a quadratic form in three variables. Find a quadratic form of the…

A: Given: g(x)= 4x22+4x32 -10x2 x3 The matrix of the…

Q: (d) Show that x² + x +4 is irreducible over Z11. (e) Write x³ + x² + x +1 as a product of irrducible…

A: An irreducible polynomial is a non-constant polynomial over a field that cannot be factored into the…

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: Show that over the ring Z/30Z[x] the polynomial f(x)=x has the factorization…

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

A: Primitive polynomial

Q: Prove that if the polynomial f(x) = x²n + a₁x²n-² + ... + an-1x² + an is divisible by g1(x) = x - 1,…

A: To Prove- Prove that if the polynomial fx = x2n + a1x2n-2 + ⋯⋯ + an-1x2 + an is divisible by g1x = x…

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

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Q: Show thet tegular a S(G) = A(G). G

A: Let us assume that graph G is regular. Now,  δ(G)= degree of the vertex with minimum degree…

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

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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: Determine whether or not the given polynomial is irreducible over Q (a) x4 - 10x² + 1 (b) (5/2)x+…

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Q: Which pairs of polynomials f, g e C[X] do have exactly one common root? O f = (X8 – 1)³, 9 = (X + X?…

A: Use the following formulae, to obtain the solution. a2-b2=a-ba+ba3-b3=a-ba2+ab+b2a3+b3=a+ba2-ab+b2