1. Which of the following is true.?Za[x]/(x +x + 1) eZ2[x]/(x + x² + x + 1)O Ck1/(x? + 1) Is a fleldx2 + 100x + 2500 n2Is irreducible over QORI1/?+1) Is a field+ 1) Is a fleld

Given: 1) a) ℤ2x/x3+x+1≅ℤ2x/x3+x2+x+1 b) ℂ[x]/x2+1 is a field. c) x2+100x+2500-n2 is irreducible…

Answered: 1. Which of the following is true.?…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Similar questions

9. Let F be a field of characteristic 2 with at least 2 elements. Then for some x,y in F (x+y)³ ‡ x³ + y³.
If A = (aij) = M3 (K) for a field K then the characteristic polynomial of A has the form p₁(x) = - det(A) + C(A)x − Tr(A)x² + x³ where the coefficient of a can be computed as C(A) = a11@22 + a22a33 + a11933 a12a21a13a31 - a23a32 (a) is it necessarily true that if A is invertible then C(A¯¹) = C(A)? (b) is it necessarily true that if det (A) = Tr(A) = 0 then A³ = -C(A)A?
Determine with explanation which of the following polynomials is irreducible over the mentioned field. 1. x° +6x4+3x<+5x-2 over Q. 2. x4+x+1 over R.
If F = (M, N, P) and all of the components of this field are polynomials in x, y, and z, then which ONE of these field operations is IMPOSSIBLE? (a) V (V · F) (b) F × (V × F) (с) F:(V x F) (d) F· (V · F) (е) F.F
10. Consider the irreducible polynomial f = X' + X³ + X² + X + 1 in Z[X]. Let a = [X] in the ring Z[X]/(f). Express (a + 1)(a² + 1) in the form aza + aza² + aa + ao where a; E Z.
Let f(x) be an irreducible polynomial over a field F. Then f(x) has a multiple root in some field extension if and only if f'(x)=0.
Let K be an extension of a field F and f(x) be a polynomial of positive degree over F, then ?∈K is a multiple root of f(x) if and only if ? is a common root of f(x) and f'(x).
Please give me correct solution for this question.
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Theorem. Let F be a field and f e F[x] a polynomial of degree n. Then there is a finite-dimensional extension of F in which ƒ factors into linear factors f(x) = (x – a1) ... (x – an). Proof. Apply the last theorem repeatedly to get getting extensions of extensions and factor out a linear factor each time until the degree is reduced to 1.

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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