1. Which of the following is true.? Z₂[x]/(x³ + x + 1) = Z₂[x]/(x³ + x² + x + 1) C[x]/(x² + 1) Is a field x² + 100x + 2500 -n² is irreducible over Q R[x]/(x² + 1) is a field

Q: Show that f(x) = x6 − 2ax³ + a is inseparable over Z3 (a), with a in any extension field of Z₂l

A: Given that                       f(x)=x6-2ax3+a We have  to show that f(x) is inseparable over Z3(a)…

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

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Q: Consider the Solow model with population growth. Assume that population can grow at two different…

A: Introduction:       The Solow Growth Model, an exogenous model of economic increase, examines…

Q: Find the splitting field x4 - x2 - 2 over Z3.

A: In mathematics, a splitting field is a field extension that contains all of the roots of a given…

Q: Let alpha be a zero of f(x) =x2 + 2x +2 in some extension field of Z3. Find the other zero of f(x)…

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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: Let F be a field of characteristic # 2. Let a and ß be roots of X2 - ae F(X) and X² − b e F(X),…

A: Since, α is a root of X2 - a = 0, we have          α2 - a = 0.  => α2 = a Since, β is a root of…

Q: Let F be a finite field with q elements and -1 -1] A 1 0 E € GL₂(F). Find N(A).

A: Definition: The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column…

Q: Show by any means that Q[r]/(x4 + 3x2 + 9x + 1) is a field.

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Q: 4. Find a monic polynomial of least degree in ℝ[x] that has 1-i and 2i as roots. write the powers of…

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Q: Find the splitting field x4 - x2 - 2 over Z3.

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Q: i. Let F be a field. Prove that a polynomial p(X) = F[X] of degree ≤ 3 is irreducible if and only if…

A: Introduction: Let F be a field. We say that a non-constant polynomial f(x) is reducible over F or…

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

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Q: Define an algebraically closed field. Show that field E is algebraically closed if and only if every…

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Q: Use the field F =Z2[x]/<x2 + x + 1> to construct a (5, 2) linearcode that will correct any…

A: Consider the given field, F=Z2x/x2+x+1 Now, abbreviate the coset a+x2+x+1 with a. Then the…

Q: (b) How many elements are there in S? Justify your answer.

A: The elements areNote: the elements of  are

Q: Show that Z[i]/<1 + i> is a field. How many elements does this fieldhave?

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Q: 22. Find all the zeros in the indicated finite field of the given polynomial with coefficients in…

A: see the calculation and answer

Q: Show that the polynomial x³-x+2 over the finite field F3 is irreducible check that, if a is any root…

A: We know that a point x=a is a root of the function fx if fa=0 i.e., if the point satisfies the…

Q: Find the splitting field E over Q, and a basis of E over Q of: (a) x*- 4 (6) x - 5

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Q: 10. Find the remainder of x4 + 3x + 2 after division by x + 3 in

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Q: For which fields F listed below is the polynomial X4 + X3 +1 e F[X] irreducible? Select all that…

A: please comment if you need any clarification. if you find my answer useful put thumbs up. Thank you.

Q: Find all values of a in Z7 such that the quotient ring Z7[x]/(p(x)) where p(x) = x³ + x² + ax + 3 is…

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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: (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: Consider the field Z3[x]/. Please explain to me how this field has 9 elements. Thank you!

A: Consider the field Z3[x]/<x2 +x+2>. Please explain to me how this field has 9 elements.

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

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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: (8) If F is a field, then it has no proper ideal. От F

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Q: Let I/Q be an extension of fields and f(X) = K[X] is an irreducible polynomial such that the degree…

A: Given:  LQ as an extension of fields and fX∈KX is an irreducible polynomial such that the degree of…

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: 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: Let α be a zero of x3 + x2 + 1 in some extension field of Z2. Solvethe equation (α + 1) x + α = α2 +…

A: Let α be a root of x3+x2+1 in some extension field of ℤ2. To Find: The solution of the equation…